Our aim in this paper is to characterize local Muckenhoupt weighted Lebesgue spaces with variable exponent by compactly supported smooth wavelets. We also investigate necessary and sufficient conditions for the corresponding modular inequalities to hold. One big achievement is that the weights with exponetial growth can be handled in the framework of variable exponents.
In this paper, we establish the sharp boundedness of p-adic multilinear Hausdorff operators on the product of Lebesgue and central Morrey spaces associated with both power weights and Muckenhoupt weights. Moreover, the boundedness for the commutators of p-adic multilinear Hausdorff operators on the such spaces with symbols in central BMO space is also obtained.
This manuscript addresses Muckenhoupt $A_{p}$ weight theory in connection to Morrey and BMO spaces. It is proved that $ω$ belongs to Muckenhoupt $A_{p}$ class, if and only if Hardy-Littlewood maximal function $M$ is bounded from weighted Lebesgue spaces $L^{p}(ω)$ to weighted Morrey spaces $M^{p}_{q}(ω)$ for $1<q< p<\infty$. As a corollary, if $M$ is (weak) bounded on $M^{p}_{q}(ω)$, then $ω\in A_{p}$. The $A_{p}$ condition also characterizes the boundedness of the Riesz transform $R_{j}$ and convolution operators $T_ε$ on weighted Morrey spaces. Finally, we show that $ω\in A_{p}$ if and only …
We define $A_{p(\cdot)}^{\rm loc}$ and show that the weighted inequality for local Hardy--Littlewood maximal operator on the Lebesgue spaces with variable exponent. This work will extend the theory of Rychkov, who developed the theory of $A_p^{\rm loc}$ weights. It will also extend the work by Cruz-Uribe. SFO, Fiorenza and Neugebaucer, who considered the Muckenhoupt class for Lebesgue spaces with variable exponents. Due to the setting of variable exponents, a new method of extension of weights will be needed; the extension method is different from the one by Rychkov. A passage to the vector-va…
Battle-Lemarie wavelet systems of natural orders are established in the paper. The main result of the work is decomposition theorem in Besov and Triebel-Lizorkin spaces with local Muckenhoupt weights, which is performed in terms of bases generated by the systems of such a type. Battle-Lemarie wavelets are splines and suit very well for the study of integration operators.
The dyadic paraproduct is bounded in weighted Lebesgue spaces $L_p(w)$ if and only if the weight $w$ belongs to the Muckenhoupt class $A_p^d$. However, the sharp bounds on the norm of the dyadic paraproduct are not known even in the simplest $L_2(w)$ case. In this paper we prove the linear bound on the norm of the dyadic paraproduct in the weighted Lebesgue space $L_2(w)$ using Bellman function techniques and extrapolate this result to the $L_p(w)$ case.
We compute the right and left democracy functions of admissible wavelet bases in variable Lebesgue spaces defined on $R^n$. As an application we give Lebesgue type inequalities for these wavelet bases. We also show that our techniques can be easily modified to prove analogous results for weighted variable Lebesgue spaces and variable exponent Triebel-Lizorkin spaces.
In the paper it is considered the generalized Faber polynomials defined inside and outside a regular curve on the complex plane. The weighted Smirnov spaces corresponding to bounded and unbounded regions are defined. It is proved that the generalized Faber polynomials forms a basis in weighted Smirnov spaces, if the weight function satisfies the Muckenhoupt condition on the regular curve. The double system of generalized Faber polynomials with complex-valued coefficients is also considered and the basis properties of such a system in weighted Lebesgue spaces over regular curves are studied.
We define a Muckenhoup-type condition on weighted Morrey spaces using the Köthe dual of the space. We show that the condition is necessary and sufficient for the boundedness of the maximal operator defined with balls centered at the origin on weighted Morrey spaces. A modified condition characterizes the weighted inequalities for the Calderón operator. We also show that the Muckenhoup-type condition is necessary and sufficient for the boundedness on weighted local Morrey spaces of the usual Hardy-Littlewood maximal operator, simplifying the previous characterization of Nakamura-Sawano-Tanaka. …
As is known, the class of weights for Morrey type spaces $\mathcal{L}^{p,\lb}(\rn) $ for which the maximal and/or singular operators are bounded, is different from the known Muckenhoupt class $A_p$ of such weights for the Lebesgue spaces $L^p(\Om)$. For instance, in the case of power weights $|x-a|^ν, \ a\in \mathbb{R}^1,$ the singular operator (Hilbert transform) is bounded in $L^p(\mathbb{R})$, if and only if $-1<ν<p-1$, while it is bounded in the Morrey space $\mathcal{L}^{p,\lb}(\mathbb{R}), 0\le \lb<1$, if and only if the exponent $\al$ runs the shifted interval $\lb-1<ν<\lb+p-1.$ A descr…
We study embeddings between generalised Besov-Morrey spaces. Both sufficient and necessary conditions for the embeddings are proved. Embeddings of the Besov-Morrey spaces into the Lebesgue spaces are also considered. Our approach requires a wavelet characterisation of the spaces which we establish for the system of Daubechies wavelets.
Let $L:=-Δ+V$ be the Schrödinger operator on $\mathbb{R}^n$ with $n\geq 3$, where $V$ is a non-negative potential which belongs to certain reverse Hölder class $RH_q(\mathbb{R}^n)$ with $q\in (n/2,\,\infty)$. In this article, the authors obtain the quantitative weighted boundedness of Littlewood--Paley functions $g_L$, $S_L$ and $g_{L,\,λ}^\ast$, associated to $L$, on weighted Lebesgue spaces $L^p(w)$, where $w$ belongs to the class of Muckenhoupt $A_p$ weights adapted to $L$.
We study the construction of nonuniform tight wavelet frames for the Lebesgue space $L^2(\mathbb{R})$, where the related translation set is not necessary a group. The main purpose of this paper is to prove the unitary extension principle (UEP) and the oblique extension principle (OEP) for construction of multi-generated nonuniform tight wavelet frames for $L^2(\mathbb{R})$. Some examples are also given to illustrate the results.
We study the continuity and compactness of embeddings for radial Besov and Triebel-Lizorkin spaces with weights in the Muckenhoupt class $A_\infty$. The main tool is a discretization in terms of an almost orthogonal wavelet expansion adapted to the radial situation.
Nowadays the theory and application of wavelet decompositions plays an important role not only for the study of function spaces (of Lebesgue, Hardy, Sobolev, Besov, Triebel-Lizorkin type) but also for its applications in signal and numerical analysis, partial differential equations and image processing. In this context it it a hard problem to construct wavelet bases for suitable function spaces on domains, e. g. the unit cube. A big step in this direction are the contributions of Hans Triebel from 2006 to 2008 where he constructed Riesz bases for classes of Besov- and Triebel-Lizorkin spaces o…
In this paper we show that the Fourier transform induces an isomorphism between the coorbit spaces defined by Feichtinger and Gröchenig of the mixed, weighted Lebesgue spaces $L_{v}^{p,q}$ with respect to the quasi-regular representation of a semi-direct product $\mathbb{R}^{d}\rtimes H$ with suitably chosen dilation group $H$, and certain decomposition spaces $\mathcal{D}\left(\mathcal{Q},L^{p},\ell_{u}^{q}\right)$ (essentially as introduced by Feichtinger and Gröbner), where the localized ,,parts`` of a function are measured in the $\mathcal{F}L^{p}$-norm. This equivalence is useful in sever…
This paper defines local weighted Hardy spaces with variable exponent. Local Hardy spaces permit atomic decomposition, which is one of the main themes in this paper. A consequence is that the atomic decomposition is obtained for the functions in the Lebesgue spaces with exponentially decaying exponent. As an application, we obtain the boundedness of singular integral operators, the Littlewood--Paley characterization and wavelet decomposition.
We prove that the operator norm of every Banach space valued Calderon-Zygmund operator T on the weighted Lebesgue-Bochner space depends linearly on the Muckenhoupt A_2 characteristic of the weight. In parallel with the proof of the real-valued case, the proof is based on pointwise dominating every Banach space valued Calderon-Zygmund operator by a series of positive dyadic shifts. In common with the real-valued case, the pointwise dyadic domination relies on Lerner's local oscillation decomposition formula, which we extend from real-valued functions to Banach space valued functions. The extens…
Let $n\ge2$ and $Ω\subset\mathbb{R}^n$ be a bounded NTA domain. In this article, the authors investigate (weighted) global gradient estimates for Dirichlet boundary value problems of second order elliptic equations of divergence form with an elliptic symmetric part and a BMO anti-symmetric part in $Ω$. More precisely, for any given $p\in(2,\infty)$, the authors prove that a weak reverse Hölder inequality with exponent $p$ implies the global $W^{1,p}$ estimate and the global weighted $W^{1,q}$ estimate, with $q\in[2,p]$ and some Muckenhoupt weights, of solutions to Dirichlet boundary value prob…
Let $T_{Ω,α}$ be the homogeneous fractional integral operator defined as \begin{equation*} T_{Ω,α}f(x):=\int_{\mathbb R^n}\frac{Ω(x-y)}{|x-y|^{n-α}}f(y)\,dy, \end{equation*} and the related fractional maximal operator $M_{Ω,α}$ is given by \begin{equation*} M_{Ω,α}f(x):=\sup_{r>0}\frac{1}{|B(x,r)|^{1-α/n}}\int_{|x-y|<r}|Ω(x-y)f(y)|\,dy. \end{equation*} In this article, we will use the idea of Hedberg to reprove that the operators $T_{Ω,α}$ and $M_{Ω,α}$ are bounded from $L^p(\mathbb R^n)$ to $L^q(\mathbb R^n)$ provided that $Ω\in L^s(\mathbf{S}^{n-1})$, $s'<p<n/α$ and $1/q=1/p-α/n$, which was …
Let $n\ge2$ and $Ω$ be a bounded Lipschitz domain in $\mathbb{R}^n$. In this article, the authors investigate global (weighted) estimates for the gradient of solutions to Robin boundary value problems of second order elliptic equations of divergence form with real-valued, bounded, measurable coefficients in $Ω$. More precisely, let $p\in(n/(n-1),\infty)$. Using a real-variable argument, the authors obtain two necessary and sufficient conditions for $W^{1,p}$ estimates of solutions to Robin boundary value problems, respectively, in terms of a weak reverse Hölder inequality with exponent $p$ or …
We study traces of weighted Triebel-Lizorkin spaces $F^s_{p,q}({\mathbb R}^n,w)$ on hyperplanes ${\mathbb R}^{n-k}$, where the weight is of Muckenhoupt type. We concentrate on the example weight $w_α(x) = |x_n|^α$ when $|x_n|\leq 1$, $x\in{\mathbb R}^n$, and $w_α(x)=1$ otherwise, where $α>-1$. Here we use some refined atomic decomposition argument as well as an appropriate wavelet representation in corresponding (unweighted) Besov spaces. The second main outcome is the description of the real interpolation space $(B^{s_1}_{p_1,p_1}({\mathbb R}^{n-k}), B^{s_2}_{p_2,p_2}({\mathbb R}^{n-k}))_{θ,r…
We study nuclear embeddings for weighted spaces of Besov and Triebel-Lizorkin type where the weight belongs to some Muckenhoupt class and is essentially of polynomial type. Here we can extend our previous results [17,19] where we studied the compactness of corresponding embeddings. The concept of nuclearity goes back to Grothendieck who defined it in [14]. Recently there is a refreshed interest to study such questions [5-8,49]. This led us to the investigation in the weighted setting. We obtain complete characterisations for the nuclearity of the corresponding embedding. Essential tools are a …
In a previous paper, we obtained several "compact versions" of Rubio de Francia's weighted extrapolation theorem, which allowed us to extrapolate the compactness of linear operators from just one space to the full range of weighted Lebesgue spaces, where these operators are bounded. In this paper, we study the extrapolation of compactness for bilinear operators in terms of bilinear Muckenhoupt weights. As applications, we easily recover and improve earlier results on the weighted compactness of commutators of bilinear Calderón-Zygmund operators, bilinear fractional integrals and bilinear Fouri…
Let $A_1$ and $A_2$ be expansive dilations, respectively, on ${\mathbb R}^n$ and ${\mathbb R}^m$. Let $\vec A\equiv(A_1, A_2)$ and $\mathcal A_p(\vec A)$ be the class of product Muckenhoupt weights on ${\mathbb R}^n\times{\mathbb R}^m$ for $p\in(1, \infty]$. When $p\in(1, \infty)$ and $w\in{\mathcal A}_p(\vec A)$, the authors characterize the weighted Lebesgue space $L^p_w({\mathbb R}^n\times{\mathbb R}^m)$ via the anisotropic Lusin-area function associated with $\vec A$. When $p\in(0, 1]$, $w\in {\mathcal A}_\infty(\vec A)$, the authors introduce the weighted anisotropic product Hardy space $…
