Zygmund-Calderon Operators in  the Weighted Variable Exponent  Spaces

Mykola Yaremenko

doi:10.47739/1019

Zygmund-Calderon Operators in the Weighted Variable Exponent Spaces

Research Article | Open Access | Volume 6 | Issue 1

Article DOI : https://doi.org/10.47739/2578-3173.mathematics.1019

Mykola Yaremenko^*

^1. National Technical University of Ukraine, Ukraine

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Corresponding Authors

Mykola Yaremenko, National Technical University of Ukraine, Ukraine

Abstract

This article is dedicated to the Zygmund-Calderon operators in the variable exponent spaces $L^{p\left ( . \right )}$ ⋅ with measurable function $p:R^{n}\rightarrow \left ( 1,\infty \right )$ .We establish that if an operator T(f )(x)= f * K with the kernel $\left | \partial _{x}^{a} K\left ( x \right )\right |\leq C\left ( a \right )\left | x \right |^{-n\left | a \right |}$ ,, $\left | a \right |\leq 1$ satisfies the $\int _{R^{n}}\left | T\left ( f \right ) \left ( x \right )\right |^{p\left ( x \right )}\omega \left ( x \right )dx\leq c_{2}\int _{R^{n}}\left | f\left ( x \right ) \right |^{p\left ( x \right )}\omega \left ( x \right )dx$ for all $\int \epsilon L^{P\left ( . \right )}\left ( R^{n} \right ),p_{s}=ess sup p\left ( x \right )< \infty$ then the weight $\omega =\frac{d\mu }{dx}$ belongs to $A_{P\left ( . \right )}$ -class. The inverse is also true, thus, if the maximal operator is bounded in $L^{p\left ( . \right )}\left ( R^{n} \right )$ and $\left | \partial _{x}^{a} K\left ( x \right )\right |\leq C\left ( a \right )\left | x \right |^{-n\left | a \right |}$ , $\left | a \right |\leq 1$ then, the inequality $\int _{R^{n}}\left | T\left ( f \right ) \left ( x \right )\right |^{p\left ( x \right )}\omega \left ( x \right )dx\leq c_{2}\int _{R^{n}}\left | f\left ( x \right ) \right |^{p\left ( x \right )}\omega \left ( x \right )dx$ holds for all $\int \epsilon L^{P\left ( . \right )}\left ( R^{n} \right )$ and each $\omega \epsilon$ $A_{P\left ( . \right )}$ .

Keywords

Harmonic Analysis; Singular Integrals; Convex seminars; Interpolation Theorem; Calderon-Zygmund Decomposition

CITATION

Yaremenko M (2024) Zygmund-Calderon Operators in the Weighted Variable Exponent Spaces. JSM Math Stat 6(1): 1019.

INTRODUCTION

The variable Lévesque spaces were introduced in 1961 by I. Tsenov who considered the problem of approximation in the Lévesque spaces [1-18]. The variable Lévesque space with a measurable function $p:R^{n}\rightarrow \left ( 1,\infty \right )$ is the set of all measurable function on $R^{n}$ the inequality $\omega =\frac{.}{dx}$ holds for some positive values of the parameterλ . The norm of the variable Lévesque space $L^{p\left ( . \right )}$ ⋅ is defined as an infimum

$\left \| f \right \|_{L^{P\left ( . \right )}}=$ inf $\left \{ \lambda > 0:\int \left ( \frac{\left | f\left ( x \right ) \right |}{\lambda } \right ) ^{p\left ( x \right )}dx< \infty \leq 1\right \}.$

The classical Lévesque spaces $L^{p\left ( . \right )}$ is a special case of ⋅ when function : $p:R^{n}\rightarrow \left ( 1,\infty \right )$ is constant.

The most prominent feature of $L^{p\left ( . \right )}$ ⋅ is existence of an analog of the Holder inequality in the weaker form $\int \left | f\left ( x \right )g\left ( x \right ) \right |dx\leq \left ( 1+\frac{1}{p_{m}} -\frac{1}{p_{s}}\right )\left \| f \right \|\left \| L^{p\left ( . \right )} \right \|\left \| g \right \|L^{p\left ( . \right )},$ where $p_{m=}$ ess inf p(x) and $p_{s}=$ ess sup P(X).

There is an essential difference between the classical Lévesque spaces and the variable $L^{p\left ( . \right )}$ ⋅ , the necessary and sufficient requirement for the operator $\tau \left ( z,f\left ( x \right ) \right )=f\left ( x-z \right )$ of translation to be bounded on p( ) L ⋅ is that the function : $p:R^{n}\rightarrow \left ( 1,\infty \right )$ be a constant. The corollary of this is that the Young lemma $\left \| f*g \right \|L^{P\left ( . \right )}\leq const \left \| f \right \|L^{p\left ( . \right )}\left \| g \right \|L^{p\left ( . \right )}$ holds for all $f\epsilon$ $L^{p\left ( . \right )}$ and all g L ∈ 1 if and only if exponential function p (⋅) is a constant.

Let M is a maximal operator then the inequality $\left \| M\left ( f \right ) \right \|_{L^{P}\left ( R^{n}, \mu \right )},\leq \sqrt[p]{A}\left \| f \right \|_{L^{p}\left ( R^{n} ,\mu \right )}$ holds for all $f\epsilon$ $L^{p\left ( . \right )}$ $\left ( R^{n}, \mu \right )$ $d\mu \left ( x \right )=\omega \left ( x \right )dx$ weightω ∈ Ap , the class Ap is characterized by inequality $\frac{1}{mes\left ( B \right )}\int _{B}d\mu \left ( x \right )\left (\frac{1}{mes\left ( B \right )}\int _{B} \omega \left ( x \right ) \tfrac{1}{1-p}dx\right )^{p-1}\leq$ A holding fork balls B.

In 2008, L. Diening and P. Hasto [6,7] generalized classes $A_{P}$ to the variable exponential Lebesgue spaces by demanding that the inequality $sup_{B}\frac{1}{\left ( mes\left ( B \right ) \right )^{p\left ( B \right )}}\left \| \omega 1_{B} \right \|\left \| L^{1} \right \|\left \| \omega ^{-1} 1_{B}\right \|\frac{q\left ( . \right )}{L^{p\left ( . \right )}}\leq A$ Holds for some constants, the minimum of these constants is the value of norm $\left \| \omega \right \|_{A_{p\left ( . \right )}}.$

Some pertinent to the subject literature reviews can be found in the L. Diening, P. Hasto works [6,7], without being complete, we present the list of some interesting research on the subject [1-25]. In this article, we consider a Zygmund-Calderon operator Ta [17] in the variable exponent spaces $L^{p\left ( . \right )}$ given in the form T(f)(x)= $\int _{R^{n}}K\left ( x-y \right )f\left ( f \right )dy$ for almost all x f ∉supp(f ) , with a singular kernel K such that, for α ≤1, the estimate $\left | \partial _{x}^{a} K\left ( x \right )\right |\leq C\left ( a \right )\left | x \right |^{-n\left | a \right |}$ holds for all n x R ∈ with the exception of x = 0 . We establish that assume f =T (f ) is a Zygmund-Calderon operator $T\left ( f \right )\left ( x \right )=f*k$ with the kernel under restriction $\left | \partial _{x}^{a} K\left ( x \right )\right |\leq C\left ( a \right )\left | x \right |^{-n\left | a \right |}$ , $\left | a \right |\leq 1$ and the $L^{p\left ( . \right )}$ -Condit $\int _{R^{n}}\left | T\left ( f \right ) \left ( x \right )\right |^{p\left ( x \right )}\omega \left ( x \right )dx\leq c_{2}\int _{R^{n}}\left | f\left ( x \right ) \right |^{p\left ( x \right )}\omega \left ( x \right )dx$ Holds for all $f\varepsilon L^{p\left ( . \right )}\left ( R^{n} \right )$ , sup ( ) n S x R p ess p (x) < ∞ , then the weight d dx µ ω = must belong to $L^{p\left ( . \right )}$ ⋅ -class. Also, we prove the inverse result, namely, presume the maximal operator is bounded in ⋅ $p:R^{n}\rightarrow \left ( 1,\infty \right )$ and operator f T f ? ( ) defined as above, then, for each weight p( ) ω ∈ A ⋅ , the inequality $\int _{R^{n}}\left | T\left ( f \right ) \left ( x \right )\right |^{p\left ( x \right )}\omega \left ( x \right )dx\leq c_{2}\int _{R^{n}}\left | f\left ( x \right ) \right |^{p\left ( x \right )}\omega \left ( x \right )dx$ Holds for all $f\varepsilon L^{p\left ( . \right )}\left ( R^{n} \right )$

Lebesgue Spaces With Variable Exponential

Let ? be an open connected subspace of the Rn . Let P(?) be a subspace of L (?) 1 such that p(.) $\varepsilon p\left ( \Omega \right )$ , $p\left ( . \right ):\Omega \rightarrow \left ( 1,\infty \right )$ .

Definition 1: For givenp(.) $\varepsilon p\left ( \Omega \right )$ we define the conjugate function $q\left ( . \right )$ by $q\left ( x \right )=\frac{p\left ( x \right )}{p\left ( x \right )-1}$ for all $x\varepsilon \Omega$ .

We denote $p_{m}\left ( \Omega ^{\tilde{~}} \right )= ess p\left ( x \right )$ and $p_{s}\left ( \Omega ^{\tilde{~}} \right )$ for fixed? ⊂ ? .

Definition 2: For fixed p(.) $\varepsilon p\left ( \Omega \right )$ , we define the functional $\rho _{p}$ By

$\rho _{p}\left ( f \right )=\int _{\Omega }\left | f\left ( x \right ) \right |^{p\left ( x \right )} dx$ (1)

FOR $f\epsilon L^{1}\left ( \Omega \right )$

Straight forward considerations yield the following properties.

Properties 1: For fixed subset n ? ⊂ R and given p P (⋅ ∈ ? ) ( ) , we have that

1) $\rho _{p}\left ( f \right )\geq 0$ For all $f\epsilon L^{1}\left ( \Omega \right )$

2) $\rho _{p}\left ( f \right )=0$ If and only if $f=0$

3) For all α β, 0 ≥ such that $\alpha +\beta$ =1, the inequality $\alpha +\beta$

$\rho _{p}\left ( \alpha f+\beta g \right )\leq \alpha \rho _{p}\left ( f \right )+\beta \rho _{p}\left ( g \right )$ (2)

$\left | f\left ( x \right ) \right |\geq \left | g\left ( x \right ) \right |$ almost everywhere and $\rho _{p}\left ( f \right )< \infty$ then $\rho _{p}\left ( f \right )\geq \rho _{p}\left ( g \right )$ and if $\rho _{p}\left ( f \right )\geq \rho _{p}\left ( g \right )$ then $\left | f\left ( x \right ) \right |\neq \left | g\left ( x \right ) \right |$ .

Definition 3: For fixed $p\left ( . \right )\epsilon p\left ( \Omega \right )$ we define a norm by $\left \| f \right \|_{L^{P\left ( . \right )}}=$ inf $\left \{ \lambda > 0:\rho _{p}\left ( \frac{f}{\lambda } \right )\leq 1\right \}$ (3)

For measurable function f. The functional space $L^{p\left ( . \right )}\left ( \Omega \right )=L^{p}\left ( \Omega \right )$ consists of all measurable functions f such that $\left \| f \right \|_{L^{P\left ( . \right )}}< \infty$

Similar to the classical Lebesgue spaces, for the $L^{p\left ( . \right )}\left ( \Omega \right )$

spaces, we can formulate an analog of the Holder norm inequality.

Theorem 2: For fixed $p\left ( . \right )\epsilon P\left ( \Omega \right )$ and conjugation functions

q (⋅) , the inequality

$\int _{\Omega }\left | f\left ( x \right ) g\left ( x \right )\right |dx\leq c_{p}\left \| f \right \|_{L^{p}}\left \| g \right \|_{L^{q}}$ (4)

With the constant $c_{p}=1+\frac{1}{p_{m}}+\frac{1}{p_{s}}$ for all $f\epsilon L^{p\left ( . \right )}\left ( \Omega \right )$ and $g\epsilon L^{P\left ( . \right )}\left ( \Omega \right )$

Proof: First, we show that $\rho _{P}\left ( \frac{f}{\left \| f \right \|}_{L^{p}} \right )=1$ holds for each $p\left ( . \right )\epsilon P\left ( \Omega \right )$ and all $f\epsilon L^{p\left ( . \right )}\left ( \Omega \right )$ , $f\neq 0$ Indeed, for all $\lambda ,0< \lambda < \left \| f \right \|_{L^{P}},$ we have

$\rho _{p}\left ( \frac{f}{\lambda } \right )\leq \left ( \frac{\left \| f \right \|_{L^{p}}}{\lambda } \right )^{p_{s}}\rho _{p}\left ( \frac{f}{\left \| f \right \|}_{L^{p}} \right ),$ Thus, there exists λ such that $\rho _{p}\left ( \frac{f}{\lambda } \right )< 1$ but $\rho _{p}\left ( \frac{f}{\left \| f \right \|_{L^{p}} } \right )\geq 1.$

Next, assuming $g\epsilon L^{p\left ( . \right )}\left ( \Omega \right )$ and $f\epsilon L^{p\left ( . \right )}\left ( \Omega \right )$ , applying the Young inequality, we estimate $\int _{\Omega }\left | \left ( \frac{f\left ( x \right )}{\left \| f \right \|_{L^{p}}} \frac{g\left ( x \right )}{\left \| g \right \|_{L^{q}}}\right ) \right |dx \leq$ $\leq \int _{\Omega }\frac{1}{p\left ( x \right )}\left | \frac{f\left ( x \right )}{\left \| f \right \|_{L^{p}}} \right |^{p\left ( x \right )} dx+\int _{\Omega }\frac{1}{q\left ( x \right )}\left \| \frac{g\left ( x \right )}{\left \| q \right \|_{L^{p}}} \right \|^{q\left ( x \right )}dx\leq \frac{1}{p_{m}}\rho _{p}\left ( \frac{f}{\left \| f \right \|_{L^{p}}} \right )+$ $\frac{1}{q_{m}}\rho _{q}\left ( \frac{g}{\left \| g \right \|_{L^{q}}} \right )\leq \left ( 1+\frac{1}{p_{m}}-\frac{1}{p_{s}} \right ),$ Since $q_{m}$ =ess inf $q\left ( x \right )=\frac{p_{s}}{p_{s}-1}$ . Thus, we obtain $\int _{\Omega }\left | f\left ( x \right ) g\left ( x \right )\right |dx \leq \left ( 1+\frac{1}{p_{m}}-\frac{1}{p_{s}} \right )\left \| f \right \|_{L^{p}}\left \| g \right \|_{L^{p}}.$

We formulate several fundamental properties of $L^{p\left ( . \right )}$ ⋅ -functions without proving them.

1) Let $f\epsilon L^{p\left ( . \right )}$ then there exists a sum-presentation of f as $f_{1}+f_{2}$ where $f_{1}\epsilon L^{p_{s}}\cap L^{p\left ( . \right )}$ and $f_{2}\epsilon L^{p_{m}}\cap L^{p\left ( . \right )}.$

2) The functional space $C_{0}^{\infty }$ is dense in $L^{p\left ( . \right )}$ , $P_{S}< \infty$

3. Assume $\left \{ f_{\left ( k \right )} \right \}\subset L^{p\left ( . \right )}$ and $lim_{k\rightarrow \infty }\left \| f_{k} -f\right \|_{L^{p\left ( . \right )}}=0,$ $f\epsilon L^{p\left ( . \right )}$ then there exists a subsequence $\left \{ f_{k\left ( t \right )}\right \}\subset L^{p\left ( . \right )} that f_{k\left ( t \right )}$ for almost everywhere.

4. Each Cauchy sequence in $L^{p\left ( . \right )}$ ⋅ converges in $L^{p\left ( . \right )}$

5. Let 1 m < p then the mapping g g ? Ψ( ) define by $\psi \left ( g ,f \right )=\int f\left ( x \right )g\left ( x \right )dx$ (5)

Is an isomorphism so that for each continuous linear functional $L^{p\left ( . \right )}$ ∗ ⋅ Ψ ∈ there exists a uniquely defined element g of $L^{p\left ( . \right )}$ ⋅ such that Ψ = Ψ( g ) and $\left \| g \right \|_{L^{g\left ( . \right )}}\approx \left \| \psi \right \|$ The space $L^{p\left ( . \right )}$ ⋅ is reflexive.

Weighted Classes $A_{p}$

First, we remind some general definitions from harmonic analysis and operator theory. A classical maximal operator M on , 2 n R n > is given by

$M\left ( f\right )\left ( x \right )=sup _{r> 0}\frac{1}{mes\left ( B\left ( r \right ) \right )}\int _{\left | y \right |< r}\left | f\left ( x-y \right ) \right |dy$ (6)

For all arbitrary locally integral functions f and all balls B r( ) of radius r > 0 in , 2 n R n > .

Let measure µ be absolutely continuous with respect to Lebesgue measure. A functional class $A_{p}$ consists of all weights $\omega \left ( x \right )=\frac{d\mu \left ( x \right )}{dx},$ which coincide with locally integral functions $\omega \epsilon L_{loc}^{1}\left ( R^{n} \right )$ , such that the estimate

$\frac{1}{mes\left ( B \right )}\int _{B}d\mu \left ( x \right)\left ( \frac{1}{mes\left ( B \right )}\int \omega \left ( x \right )^{\frac{1}{1-p}} dx\right )^{p-1}\leq A$ (7)

Holds for all balls B, where p+q = pq, $p\epsilon \left ( 1,\infty \right ) .$ $p\epsilon \left ( 1,\infty \right ) .$

The $A_{p}$ bound of the weight ω is a minimal constant for which (2) holds.

Applying the Holder inequality, we can prove the following lemma.

Lemma 1: For the weight ω to belong to $A_{p}$ -class it is necessary and sufficient that the estimate

$\frac{1}{mes\left ( B \right )}\int _{B}\left | f\left ( x \right ) \right |dx\leq c\left ( \frac{1}{\mu \left ( B \right )}\int _{B} \left | f\left ( x \right ) \right |^{p}d\mu \left ( x \right )\right )^{\frac{1}{p}}$ (8)

Holds for all $f\epsilon L_{loc}^{1}\left ( R^{n} \right )$ and all balls B, where ω µ ( x )dx =d $\mu$ ) = (x ).

Definition 4: The functional class BMO (bounded mean oscillation) consists of all locally integral functions f such that the inequality

$\frac{1}{mes\left ( B \right )}\int _{B}\left | f\left ( x \right ) -\frac{1}{mes\left ( B \right )}\int _{B}f\left ( x \right )dx\right |dx\leq A$ (9)

Holds for all balls B.

An important property of $A_{p}$ - weights is given by the next theorem.

Theorem 3: Let ω ∈ $A_{p}$ then the inequality

$\int _{B^{n}}\left ( M\left ( f \right )\left ( x \right ) \right )^{p}\omega \left ( x \right )dx\leq A\int _{B^{n}}\left | f\left ( x \right ) \right |^{p}\omega \left ( x \right )dx$ (10)

Holds for all $\int \epsilon L^{P}\left ( d\mu \right )$ and for each $p\epsilon \left ( 1,\infty \right ).$

Now, we can generalize the definition of $A_{p}$ - class to functional spaces $L^{p\left ( . \right )}\left ( \Omega \right ),p\left ( . \right )\epsilon p\left ( \Omega \right ).$

Definition 5: For given $p\left ( . \right )\epsilon p\left ( \Omega \right ).$ a weight $\omega \left ( x \right )=\frac{d\mu \left ( x \right )}{dx},$ belongs to the variable class $A_{p\left ( . \right )}$ ⋅ if the inequality

$sup_{B}\frac{1}{\left ( mes\left ( B \right ) \right )^{p\left ( B \right )}}\left \| \omega 1_{B} \right \|\left \| \omega ^{-1} 1_{B}\right \|_{L^{\frac{q\left ( . \right )}{p\left ( . \right )}}}\leq A$ (11)

Holds for all balls B clos ⊂ ?( ), and some constants A , where $p\left ( B \right )=\left ( \frac{1}{mes\left ( B \right )} \int _{B}p\left ( x \right )^{-1}dx\right )^{-1}$ (12)

and $p\left ( x\right ) ^{-1}+q\left ( x \right )^{-1}=1$

Theorem 4: Let p P (⋅) ∈ (? ) then a weight $\omega \left ( x \right )=\frac{d\mu \left ( x \right )}{dx},$ belongs to the variable $A_{p\left ( . \right )}$ ⋅ - class if and only if the inequality

$\left ( \frac{1}{mes\left ( B \right )}\int _{B}\left | f\left ( x \right ) \right |dx \right )^{p\left ( B \right )} \leq C_{1}\left ( \mu \left ( B \right ) \right )^{-1}\int _{B}f\left ( x \right )^{p\left ( x \right )}\omega \left ( x \right )dx$ (13)

Holds for all nonnegative $f\epsilon L^{p\left ( . \right )}\left ( d\mu \right )$ and all balls B.

Proof: Let $\omega \epsilon A_{p\left ( . \right )}$ ⋅ and applying the Holder inequality for variable Lebesgue spaces, we obtain

$\left ( \frac{1}{mes\left ( B \right )}\int _{B}\left | f\left ( x \right ) \right |dx \right )^{p\left ( B \right )}=$ $\left ( \frac{1}{mes\left ( B \right )}\int _{B}\left | f\left ( x \right ) \right |dx \right )^{p\left ( B \right )}=\left ( \frac{1}{mes\left ( B \right )}\int _{B} f\left ( x \right )\omega \left ( x \right )^{\frac{1}{p\left ( x \right )}}\omega \left ( x \right )^{-\frac{1}{p\left ( x \right )}}dx\right )^{p\left ( B \right )}\leq$

$c\frac{1}{mes\left ( B \right )p\left ( B \right )}\left ( \int _{B}\left ( f\left ( x \right ) \right )^{p\left ( x \right )} \omega \left ( x \right )dx\right )\left \| \omega ^{-1} \right \|_{L^{\frac{Q\left ( . \right )}{P\left ( . \right )}}},$ Applying the definition of $A_{p\left ( . \right )}$ ⋅ -class, we deduce the first statement of the theorem.

Conversely, we take f (x)= $\left ( \omega \left ( x \right )+\varepsilon \right )^{-\frac{q\left ( x \right )}{p\left ( x \right )}}$ and have

$\frac{1}{mes\left ( B \right )p\left ( B \right )}\left \| \omega 1_{B} \right \|_{L^{1}}\left \| \left ( \omega +\varepsilon \right )^{-1}1_{B} \right \|_{L^{\frac{q\left ( x \right )}{p\left ( x \right )}}}\leq c_{1}$

Take the limit as ε → 0 and obtain that ω ∈ $A_{p\left ( . \right )}$

For the weighted variable Lebesgue space $L^{p\left ( . \right )}\left ( R^{n}, \mu \right )$ ⋅ , one can prove an analog of Theorem 3 as follows.

Theorem (analog of theorem 3) 5: For fixed P (.) ∈p (R) ⋅ ∈ and weight ω∈ $A_{p\left ( . \right )}$ ⋅ , if maximal operator M is continuous on $L^{q\left ( . \right )}\left ( d\mu \right )$ then the inequality

$\int _{B^{n}}\left ( M\left ( f \right )\left ( x \right ) \right )^{p\left ( x \right )}\omega \left ( x \right )dx\leq A\int _{B^{n}}\left | f\left ( x \right ) \right |^{p\left ( x \right )}\omega \left ( x \right )dx$ (14)

Holds for all $f\epsilon L^{p\left ( . \right )}\left ( d\mu \right )$ and for some constants A

Pseudo-Differential Operators

The singular integral realization of the pseudo-differential operator $T_{0}$ can be present as

$T_{0}\left ( f \right )\left ( x \right )=\int _{R^{n}}k\left ( x,y \right )f\left ( x -y \right )dy$ (15)

Or in classical form

$Description: T_{a}\left ( f \right )\left ( x \right )=\int _{R^{n}}K\left ( x,y \right )f\left ( y \right )dy$ (16)

For almost all $Description: x\euro$ supp $Description: \left ( f \right )$ , where $Description: K\left ( x,y \right )=k\left ( x,x-y \right )$ and the Fourier transform

$Description: a\left ( x,\xi \right )=\int _{R^{n}}exp\left ( -2\pi y.\xi \right )k\left ( x,y \right )dy.$ (17)

We consider a convolution operator in the form

$Description: T\left ( f \right )\left ( x \right )=\int _{R^{n}}K\left ( x-y \right )f\left ( y \right )dy$ (18)

For almost all $Description: x\euro supp\left ( f \right ).$ The natural assumption on the integral kernel K is that there exists a smooth function K(x), for all $Description: x\euro R^{n}$ except x ≠ 0 , such that the kernel agrees with K(x) on elements of $Description: C^{\infty }_{o}\left ( R^{n} \right ),$ which vanish on the neighborhood of x = 0, for all $Description: \left | a \right |\leq 1$ we assume

$\left | \partial _{x}^{a} K\left ( x \right )\right |\leq C\left ( a \right )\left | x \right |^{-n\left | a \right |}$ (19)

For all $Description: x\, \in \, R^{n}$ except origin. This class of pseudo-differential operators satisfies the Zygmund-Calderon conditions.

Theorem 6: Let $Description: p\left ( \cdot \right )\in P\left ( R^{n} \right )$ and let $Description: f \mapsto T\left ( f \right )$ be a convolution operator $Description: T\left ( f \right )\left ( x \right )=f*k$ corresponding to kernel K such that If the integral inequality (21)

Holds for all $Description: f\in L^{p\left ( \cdot \right )}\left ( R^{n} \right )$ then the weight $Description: f\in L^{p\left ( \cdot \right )}\left ( R^{n} \right )\omega \left ( x \right )=\frac{d\mu \left ( x \right )}{dx}$ satisfies the inequality (22)

Namely $Description: \omega \in A_{p\left ( \cdot \right )}.$

Proof: Applying conditions on the kernel, we have

$Description: \left | K\left ( x+z \right )-K\left ( x \right ) \right |\leq \breve{c}\left | x \right |^{-n}$

For $Description: \left | z \right |\leq \breve{c}\left | x \right |.$ Assume $Description: B\left ( \tilde{x} ,\rho \right )$ then we have that the inequality

Holds for all $Description: x\in B\left ( \tilde{x} +\rho\hat{x} ,\rho \right ),$ the application of the integral inequality yields

Taking $Description: B\left ( \tilde{x} +\rho \hat{x},\rho \right )$ instead of $Description: B\left ( x,\rho \right ),$ we obtain

Theorem 7: Let $Description: p\left ( \cdot \right )\in P\left ( R^{n} \right )$ such that $Description: p_{s}=ess\underset{x\in R^{n}}{}sup \, p\left ( x \right )< \infty ,$ and let $Description: f \mapsto T\left ( f \right )$ be a convolution operator corresponding to kernel K

under the assumption $\left | \partial _{x}^{a} K\left ( x \right )\right |\leq C\left ( a \right )\left | x \right |^{-n\left | a \right |}$ , $\left | a \right |\leq 1$

by T (f )(x)= f *K. Assume that the maximal operator is bounded in , then, for each weight ω ∈ $A_{p\left ( . \right )}$ ⋅ , the inequality

$\int _{B^{n}}\left ( T\left ( f \right )\left ( x \right ) \right )^{p\left ( x \right )}\omega \left ( x \right )dx\leq A\int _{B^{n}}\left | f\left ( x \right ) \right |^{p\left ( x \right )}\omega \left ( x \right )dx$ (23)

Holds for all $f\epsilon L^{p\left ( . \right )}\left ( R^{n} \right )$

Proof: Let T , ε >0 be a truncated approximation with kernel $K_{\epsilon }\left ( x \right )=K\left ( x \right )1\left ( \left \{ \left | x \right |\geq \varepsilon \right \} \right )$ so that

$Description: T\left ( f \right )\left ( x \right )=\int _{R^{n}}K\left ( x-y \right )f\left ( y \right )dy$

And we define $T_{*}\left ( f \right )\left ( x \right )=sup_{\varepsilon }\left | T\varepsilon \left ( f \right ) \left ( x \right )\right |.$

We show that the inequality

mes $\left ( \left \{ x:T_{*} \left ( f \right )\left ( x \right )> a,f\left ( x \right )\leq c\tilde{a}\right \} \right )\leq$

$c_{2}c\tilde{}\left ( 1-b\breve{} \right )^{-1} mes \left ( \left \{ x:T_{*}\left ( f \right ) \left ( x \right )\right > b\tilde{}a\tilde{}\} \right )$

Holds for all $f\varepsilon C_{0}^{\infty }$ and for all b <1 a > 0, and c > 0 .

Indeed, for fixed ε > 0 and for all $f\varepsilon C_{0}^{\infty }$ is a continuous function, therefore, there exists an open coverage $\Theta$ such that $T_{*}\left ( f \right )\left ( x \right )=sup_{\varepsilon }\left | T_{\varepsilon } \left ( f \right )\left ( x \right )\right |> b\tilde{a}$ is open, this open coverage Θ is decomposed into a disjoint union of Whitney cubes UQi

Now, we decompose the function f into the sum $f_{1}+f_{2}$ of two functions $f_{1}=f_{1_{B}}$ AND $f_{1}=f_{1_{R^{n/B}}}$ For $b_{1}\breve{}+b_{2}^{\sim }$ =1 we obtain $\left \{ x:T_{*} \left ( f \right )\left ( x \right )> a\right \}\subset \left \{ x:T_{*} \left ( f_{1} \right )\left ( x \right )> b^{\sim }a\right \}\cup \left \{ x:T_{*} \left ( f_{2} \right )\left ( x \right )> b^{\sim }_{1}a\right \}$ Since for all $f\varepsilon L^{1}\left ( R^{n} \right )$ there is an inequality

mes $\left \{ x:T_{*} \left ( f \right )\left ( x \right )> a\right \}\leq \frac{c_{2}}{a}\int _{R^{n}}\left | f\left ( x \right ) \right |dx,$

We have

mes $\left \{ x\epsilon Q_{i}:T_{*}\left ( f_{1} \right )\left ( x \right )\right> > b_{1}^{\sim }a \}\leq \frac{c_{2}}{\alpha b^{\sim }_{1}}\int _{R^{n}}\left | f\left ( x \right ) \right |dx$

and

$\int _{R^{n}}\left | f\left ( x \right ) \right |dx\leq c^{\sim }c_{2}\alpha mes \left ( Q_{i} \right )$

Thus, we obtain

$\left \{ x:T_{*} \left ( f \right )\left ( x \right )> a\right \}\leq \frac{c_{2}}{b_{1}}$ $mes \left ( Q_{i} \right )$

Next, we must estimate the f 2 -term. Applying our conditions, we calculate

$\int _{\left | y-z \right |\geq s}\frac{\left | f\left ( y \right ) \right |}{\left | y-z \right |^{n+1}}dy=\int _{\left | y \right |\geq s}\frac{/f\left ( y-z \right )/}{\left | y \right |^{n+1}}dy=$

$=\sum _{k=0,1..2^{i}}\int __{}s\leq {\left \| y \right \|\leq 2^{i+1}}\frac{/f\left ( y-z \right )/}{\left | y \right |^{n+1}}dy\leq 2c_{1}^{\sim }f\left ( z \right )$

Since $\left | K_{\varepsilon }\left ( x^{\sim }-y \right ) -K_{\varepsilon }\left ( x-y \right )\right |\leq \frac{c_{2}s}{\left | y-z \right |^{n+1}}$ for $x\epsilon Q_{i,} y\epsilon R^{n}/B$ and $R^{n}/B\subset \left \{ y: \left | y-z \right |\geq s\right \},$ the ball B has center at x . Therefore, we estimate

$\left | T_{\epsilon }\left ( f_{2} \right )\left ( x^{\sim } \right )-T_{\epsilon } \left ( f_{2} \right )\left ( x \right )\right |\leq c_{2}f\left ( z \right )$

For all $x\epsilon Q_{i,}$ Taking the supreme over all ε > 0 , we have

$T_{*}\left ( f_{2} \right )\left ( x \right )\leq T_{*} \left ( f_{2} \right )\left ( x \right )+c_{2}f\left ( z \right )\leq \alpha b^{\sim }$

For all $x\epsilon Q_{i,}$

So, we choose $b_{2}^{\sim }\geq b^{\sim }+c_{2}c$ and $b_{1}^{\sim }+ b_{2}^{\sim }=1$ then we have

mes $\left ( \left \{ x:T_{*} \left ( f \right )\left ( x \right )> \alpha , f\left ( x \right )\leq c^{\sim }\alpha \right \} \right )\leq$

$\leq c_{2}cb^{\sim }_{1} mes \left ( Q^{i} \right )$ ,

if $c_{2}c^{\sim }\left ( 1-b^{\sim } \right )\geq 2^{-1}$ then we choose new $c_{2}$ as $2c_{2}$ and obtain

mes $\left ( \left \{ x:T_{*} \left ( f \right )\left ( x \right )> \alpha , f\left ( x \right )\leq c^{\sim }\alpha \right \} \right )\leq$

$\leq c_{2}c^{\sim } \left ( 1-b \right )^{-1} mes \left ( Q^{i} \right ).$

Taking the sum over all cubesQi , we obtain

mes $\left ( \left \{ x:T_{*} \left ( f \right )\left ( x \right )> \alpha , f\left ( x \right )\leq c^{\sim }\alpha \right \} \right )\leq$

$\leq c_{2}c^{\sim } \left ( 1-b \right )^{-1}$ mes $\left ( \left \{ x:T_{*}\left ( f \right )\left ( x \right )\geq b^{\sim } a\right \} \right )$

For all α > 0 and each 0 1 < < b and each0 < c.

Next, we show that the inequality

$\mu \left ( \left \{ x:T_{*} \left ( f \right )\left ( x \right )> \alpha , f\left ( x \right )\leq c^{\sim }\alpha \right \} \right )\leq$

$\leq \alpha \mu \left ( \left \{ x:T_{*}\left ( f \right )\left ( x \right )\geq b^{\sim } a\right \} \right )$

Holds for all $Description: f\in C_{o}^{\infty }$ and for all $Description: \tilde{b}< 1,\, \alpha > 0$ and some $Description: \tilde{a}< 1,\, \tilde{c}> 0,$ constant $Description: \tilde{a}$ depends on the weight function.

Indeed, in previous consideration, we fix $Description: \tilde{b}< 1$ and choose $Description: \tilde{c}$ so that $Description: c_{2}\tilde{c}\left ( 1-\tilde{b} \right )^{-1}$ is small enough, then the inequality

$Description: \leq mes\left ( Q_{i} \right ).$

Holds for some small enough positive $Description: \delta$ and all cubes.

Assuming $Description: \tilde{a}=\delta < 1,$ we summate over all cubes and obtain

$Description: \leq \delta \mu \left ( Q \right ).$

We will need the following properties of the $Description: p\left ( \cdot \right )$ -spaces. For constant $Description: \tilde{a}< 1,$ we choose $Description: \tilde{b}< 1,$ such that the inequality $Description: \tilde{a}< \tilde{b}^{ps}$ holds for all Description: x. Let Description: f and Description: g be a nonnegative function such that inequality

$Description: \mu \left ( \left \{ x:f\left ( x \right ) > \alpha ,g\left ( x \right )\leq \tilde{c}\alpha \right \} \right )\leq$

$Description: \leq \tilde{a}\mu \left ( \left \{ x:f\left ( x \right ) \tilde{b}\alpha \right \} \right )$

Holds for all $Description: \alpha > 0.$ Then, the inequality

Holds with some constants $Description: c_{3}$ and under the condition $Description: \tilde{a}< \tilde{b}^{ps}$ and $Description: f\in L^{p\left ( \cdot \right )}$ Description: . Proving of this statement is similar to standard one.

This proves our statement for f $Description: f\in C^{\infty }_{o}$ since $Description: \left | T_{*}\left ( f \right )\left ( x \right ) \right |\leq \left ( 1+\left | x \right | \right )^{-n}$ holds for all $Description: f\in C^{\infty }_{o}.$

The extension to the whole $Description: L^{p\left ( \cdot \right )}\left ( R^{n} ,\mu \right )$ follows from the standard argument that each element of $Description: L^{p\left ( \cdot \right )}\left ( R^{n} ,\mu \right )$ can be approximated by ele elements of $Description: C_{0}^{\infty }\left ( R^{n} \right ).$

JSM Mathematics and Statistics

JSM Mathematics and Statistics

JSM Mathematics and Statistics

Zygmund-Calderon Operators in the Weighted Variable Exponent Spaces

Abstract

Keywords

CITATION

INTRODUCTION

REFERENCES

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JSciMed

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Zygmund-Calderon Operators in the Weighted Variable Exponent Spaces

Abstract

Keywords

CITATION

INTRODUCTION

REFERENCES

Journals

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