Piotr Hajlasz, Zhuomin Liu
In this note we study boundedness of a large class of maximal operators in Sobolev spaces that includes the spherical maximal operator. We also study the size of the set of Lebesgue points with respect to convergence associated with such maximal operators.
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We generalize in this article the classical Sobolev's and Sobolev's trace inequalities on the Grand Lebesgue Spaces under monomial weight instead the classical Lebesgue or grand Lebesgue Spaces. We will distinguish the classical Sobolev's inequality and the so-called trace Sobolev's inequality.
We extend in this article the classical imbedding theorems for fractional Lebesgue-Sobolev's spaces into the so-called Grand Lebesgue spaces, with sharp constant evaluation.
We extend the classical Hardy-Sobolev-Poincare-Wirtinger inequalities from the ordinary Lebesgue-Riesz spaces into the Grand Lebesgue ones, with exact constants evaluation.
We study in this short preprint the theory of trigonometric approximation in the so-called Banach functional rearrangement invariant Sobolev-Grand Lebesgue Spaces.
We extend in this article the classical Sobolev inequalities for the module of continuity for the functions belonging to the integer order Sobolev's space on the Sobolev-Bilateral Grand Lebesgue spaces. As a consequence, we deduce the fractional Orlicz-Sobolev imbedding theorems and investigate the rectangle module of continuity of non-Gaussian multiparameter random fields.
In this short article we generalize the Sobolev's inequalities for the module of continuity for the functions belonging to the classical Lebesgue space on the (Bilateral) Grand Lebesgue spaces. We construct also some examples in order to show the exactness of obtained results.
We give in this short paper the exact value for norms of two operators of Hardy-Sobolev type acting between two weight Grand Lebesgue Space (GLS) based on the whole multidimensional Euclidean space.
We deduce an extension theorem for the so-called Sobolev-Grand Lebesgue Spaces defined on the suitable subsets of the whole finite-dimensional Euclidean space, and estimate the norms of correspondent extension operator, which may be choosed as linear.
Stability of weighted composition strongly continuous semigroups acting on Lebesgue and Sobolev spaces is studied, without the use of spectral conditions on the generator of the semigroup. Applications to the generalized von Foerster - Lasota semigroup and a comparison with hypercylicity conditions are presented.\
Wiener-Lebesgue point property for Sobolev Functions on Metric Spaces
本文为支持Poincaré不等式的完备倍测度度量空间上的一阶Sobolev函数建立了Wiener型积分条件，该条件比Lebesgue点性质更强，仅非Lebesgue点集容量有微小增加。
In this short article we show a particular version of the Hedberg inequality which can be used to derive, in a very simple manner, functional inequalities involving Sobolev and Besov spaces in the general setting of Lebesgue spaces of variable exponents and in the framework of Orlicz spaces.
In this article, we show that a function $f\in M^{s,p}(X),$ $0<s\leq 1,$ $0<p<1,$ where $X$ is a doubling metric measure space, has generalized Lebesgue points outside a set of $\mathcal{H}^h$-Hausdorff measure zero for a suitable gauge function $h.$
In this article we generalize the classical Sobolev's and Sobolev's trace inequalities on the Grand Lebesgue Spaces instead the classical Lebesgue Spaces. We will distinguish the classical Sobolev's inequality and the so-called trace Sobolev's inequality. We consider for simplicity only the case of whole space.
We give a simple characterization of chaos for weighted composition $C_0$-semigroups on $L^p_ρ(Ω)$ for an open interval $Ω\subseteq\mathbb{R}$. Moreover, we characterize chaos for these classes of $C_0$-semigroups on the closed subspace $W^{1,p}_*(Ω)$ of the Sobolev space $W^{1,p}(Ω)$ for a bounded interval $Ω\subset\mathbb{R}$. These characterizations simplify previously obtained characterization of chaos for these classes of $C_0$-semigroups.
We approximate functions defined on smooth bounded domains by elements of the eigenspaces of the Laplacian or the Stokes operator in such a way that the approximations are bounded and converge in both Sobolev and Lebesgue spaces. We prove an abstract result referred to fractional power spaces of positive, self-adjoint, compact-inverse operators on Hilbert spaces, and then obtain our main result by using the explicit form of these fractional power spaces for the Dirichlet Laplacian and Stokes operators. As a simple application, we prove that all weak solutions of the incompressible convective B…
For $μ>β>0$, the generalized Stieltjes operators $$ \mathcal{S}_{β,μ} f(t):={t^{μ-β}}\int_0^\infty {s^{β-1}\over (s+t)^μ}f(s)ds, \qquad t>0, $$ defined on Sobolev spaces $\mathcal{T}_p^{(α)}(t^α)$ (where $α\ge 0$ is the fractional order of derivation and these spaces are embedded in $L^p(\RR^+)$ for $p\ge 1$) are studied in detail. If $0 < β- \pp < μ$, then operators $\mathcal{S}_{β,μ}$ are bounded (and we compute their operator norms which depend on $p$); commute and factorize with generalized Cesáro operator on $\mathcal{T}_p^{(α)}(t^α)$ . We calculate and represent explicitly their spectrum…
The structure of non-compactness of optimal Sobolev embeddings of $m$-th order into the class of Lebesgue spaces and into that of all rearrangement-invariant function spaces is quantitatively studied. Sharp two-sided estimates of Bernstein numbers of such embeddings are obtained. It is shown that, whereas the optimal Sobolev embedding within the class of Lebesgue spaces is finitely strictly singular, the optimal Sobolev embedding in the class of all rearrangement-invariant function spaces is not even strictly singular.
We derive in this preprint the exact up to multiplicative constant non-asymptotical estimates for the norms of some non-linear in general case operators, for example, the so-called maximal functional operators, in two probabilistic rearrangement invariant norm: exponential Orlicz and Grand Lebesgue Spaces. We will use also the theory of the so-called Grand Lebesgue Spaces (GLS) of measurable functions.
Two-weight norm estimates for the double Hardy transforms and strong fractional maximal functions are established in variable exponent Lebesgue spaces. Derived conditions are simultaneously necessary and sufficient in the case when the exponent of the right-hand side space is constant.
We obtain some new characterizations of a variable version of Lipschitz spaces in terms of the boundedness of commutators of sharp maximal functions, fractional maximal functions or fractional maximal commutators in the context of the variable Lebesgue spaces, where the symbols of the commutators belong to the variable Lipschitz space. A useful tool is that a symbol $b$ belongs a variable Lipschitz space of pointwise type if and only if $b$ belongs to a variable Lipschitz space of integral type.
Non-Euclidean unification of isoperimetric profiles and grand Lebesgue-Sobolev scales
本文在完备可分度量测度空间上，通过显式变换统一等周轮廓和Grand Lebesgue-Sobolev尺度，建立嵌入定理及其逆定理，并在Heisenberg群等非欧几里得例子中验证。
This paper is a continuation of the papers [21] and [22]. Here we shall investigate the asymptotic behaviour of Weyl and Bernstein numbers of embeddings of Sobolev spaces with dominating mixed smoothness into Lebesgue spaces.
We present a simple proof of the continuity, in the sense distributions, of the minors of the differential matrices of mappings belonging to grand Sobolev spaces. Such function spaces were introduced in connection with a problem on minimal integrability of the Jacobian and are useful in certain aspects of geometric function theory and partial differential equations.
Sobolev extensions, interpolation inequalities and consequences
本文引入Lebesgue W^{1,p}扩展域概念，证明任何W^{1,p}扩展域（1<p<∞）都是Lebesgue W^{1,p}扩展域，并推导出类似全空间形式的Sobolev插值不等式，讨论其应用。
We introduce and study fractional variable exponents Sobolev trace spaces on any open set in the Euclidean space equipped with the Lebesgue measure. We show that every equivalence class of Sobolev functions has a quasicontinuous representatives. We use the relative capacity to characterize completely the zero trace fractional variable exponents Sobolev spaces. We also give a relative capacity criterium for removable sets.
We study the Hardy-Littlewood maximal operator in the Musielak-Orlicz-Sobolev space $W^{1,φ}(\mathbb{R}^n)$. Under some natural assumptions on $φ$ we show that the maximal function is bounded and continuous in $W^{1,φ}(\mathbb{R}^n)$.
It has been known that sharp Sobolev embeddings into weak Lebesgue spaces are non-compact but the question of whether the measure of non-compactness of such an embedding equals to its operator norm constituted a well-known open problem. The existing theory suggested an argument that would possibly solve the problem should the target norms be disjointly superadditive, but the question of disjoint superadditivity of spaces $L^{p,\infty}$ has been open, too. In this paper, we solve both these problems. We first show that weak Lebesgue spaces are never disjointly superadditive, so the suggested te…
Precompact Sets in Matrix Weighted Lebesgue Spaces with Variable Exponent
本文研究了变指数矩阵加权 Lebesgue 空间中的紧致性，分别利用平移算子、平均算子和近似恒等给出了相应的判据，并进一步探讨了变指数矩阵加权 Sobolev 空间中的紧致性。
In the paper two-weighted norm estimates with general weights for Hardy-type transforms, maximal functions, potentials and Calderón-Zygmund singular integrals in variable exponent Lebesgue spaces defined on quasimetric measure spaces $(X, d, μ)$ are established. In particular, we derive integral-type easily verifiable sufficient conditions governing two-weight inequalities for these operators. If exponents of Lebesgue spaces are constants, then most of the derived conditions are simultaneously necessary and sufficient for appropriate inequalities. Examples of weights governing the boundedness …
We characterize a weighted norm inequality which corresponds to the embedding of a class of absolutely continuous functions into the fractional order Sobolev space. The auxiliary result of the paper is of independent interest. It comprises of several types of necessary and sufficient conditions for the boundedness of the Hardy--Steklov operator (an integral operator with two variable boundaries of integration) in weighted Lebesgue spaces.
The Riemann Mapping Theorem states existence of a conformal homeomorphism $φ$ of a simply connected plane domain $Ω\subset\mathbb C$ with non-empty boundary onto the unit disc $\mathbb D\subset \mathbb C$. In the first part of the paper we study embeddings of Sobolev spaces $\overset{\circ}{W_{p}^{1}}(Ω)$ into weighted Lebesgue spaces $L_{q}(Ω,h)$ with an {}"universal" weight that is Jacobian of $φ$ i.e. $h(z):=J(z,φ)=| φ'(z)|^2$. Weighted Lebesgue spaces with such weights depend only on a conformal structure of $Ω$. By this reason we call the weights $h(z)$ conformal weights. In the second pa…
The present paper is devoted to a theory of profile decomposition for bounded sequences in \emph{homogeneous} Sobolev spaces, and it enables us to analyze the lack of compactness of bounded sequences. For every bounded sequence in homogeneous Sobolev spaces, the sequence is asymptotically decomposed into the sum of profiles with dilations and translations and a double suffixed residual term. One gets an energy decomposition in the homogeneous Sobolev norm. The residual term becomes arbitrarily small in the critical Lebesgue or Sobolev spaces of lower order, and then, the results of decompositi…
For kernels $ν$ which are positive and integrable we show that the operator $g\mapsto J_νg=\int_0^x ν(x-s)g(s)ds$ on a finite time interval enjoys a regularizing effect when applied to Hölder continuous and Lebesgue functions and a "contractive" effect when applied to Sobolev functions. For Hölder continuous functions, we establish that the improvement of the regularity of the modulus of continuity is given by the integral of the kernel, namely by the factor $N(x)=\int_0^x ν(s)ds$. For functions in Lebesgue spaces, we prove that an improvement always exists, and it can be expressed in terms of…
Sobolev regularity for a class of local fractional new maximal operators
本文研究局部新极大算子及其分数版本的正则性，给出逐点梯度估计，证明其在Sobolev空间上的有界性，并推广到零边值Sobolev空间。
Equivalence of Sobolev norms in Lebesgue spaces for Hardy operators in a half-space
本文研究了半空间中含势函数的哈代算子生成的Sobolev空间尺度等价性，利用新的平方函数估计方法，扩展了L^2结果并覆盖了吸引势情况。
Here we obtain order estimates for widths of weighted Sobolev classes in the weighted Lebesgue space where parameters of the second weight satisfy some limiting conditions.
We prove sufficient conditions for the boundedness of the maximal operator on variable Lebesgue spaces with weights $φ_{t,γ}(τ)=|(τ-t)^γ|$, where $γ$ is a complex number, over arbitrary Carleson curves. If the curve has different spirality indices at the point $t$ and $γ$ is not real, then $φ_{t,γ}$ is an oscillating weight lying beyond the class of radial oscillating weights considered recently by V. Kokilashvili, N. Samko, and S. Samko.
The maximal operator on variable Lebesgue spaces: an ${\mathcal A}_{\infty}$-characterization
本文通过可变指数$A_{\infty}$条件，给出了最大算子在可变指数空间上有界的新的判据。
We derive criteria governing two-weight estimates for multilinear fractional integrals and appropriate maximal functions. The two and one weight problems for multi(sub)linear strong fractional maximal operators are also studied; in particular, we derive necessary and sufficient conditions guaranteeing the trace type inequality for this operator. We also establish the Fefferman-Stein type inequality, and obtain one-weight criteria when a weight function is of product type. As a consequence, appropriate results for multilinear Riesz potential operator with product kernels follow.
An apparently new concept of maximal mean difference quotient is defined for functions in the Lebesgue space $L_{loc}(R^n)$. Our definitions are meaningful for vector valued functions on general measure metric spaces as well and seem to lead to the most natural class of metric Sobolev spaces. The discussion of higher order Sobolev spaces and higher order mean difference quotients on regular subsets of Euclidean spaces is also possible in the context of the generalized Taylor-Whitney jets. This paper is a direct sequel to the papers: B. Bojarski, Taylor expansions and Sobolev spaces, Bull. Geor…
This paper concerns the following question: given a subset $E$ of $\mathbb{R}^n$ with empty interior and an integrability parameter $1<p<\infty$, what is the maximal regularity $s\in\mathbb{R}$ for which there exists a non-zero distribution in the Bessel potential Sobolev space $H^{s,p}(\mathbb{R}^n)$ that is supported in $E$? For sets of zero Lebesgue measure we apply well-known results on set capacities from potential theory to characterise the maximal regularity in terms of the Hausdorff dimension of $E$, sharpening previous results. Furthermore, we provide a full classification of all poss…
We introduce a notion of maximal potentials and we prove that they form bounded operators from $L^p$ to the homogeneous Sobolev space $\dot{W}^{1,p}$ for all $n/(n-1)<p<n$. We apply this result to the problem of boundedness of the spherical maximal operator in Sobolev spaces.
Lebesgue space estimates for spherical maximal functions on Heisenberg groups
本研究证明海森堡群上余维2球面伸缩对应的局部极大算子的$L^p\ o L^q$估计（除两端点外最优），可改进全局极大算子稀疏支配界，还探讨了缺项变体及向Métivier群的推广。
By the Hardy-Littlewood-Sobolev theorem the classical Riesz potential is bounded on Lebesgue spaces. E. Nakai and H. Sumitomo [16] extended that theorem to the Orlicz spaces. We introduce generalized potential operators on commutative hypergroups and under some assumptions on the kernel we showed the boundedness of these operators from Lebesgue space into certain Orlicz space. Our result is an analogue of Theorem 1.3 in [16].
We investigate the class $\mathcal{B}^{loc}(\mathbb{R}^{n})$ of exponents $p(\cdot)$ for with local Hardy-Littlewood maximal operator is bounded in $L^{p(\cdot)}(\mathbb{R}^{n})$ space. Littlewood-Paley square-function characterization of $L^{p(\cdot)}(\mathbb{R}^{n})$ spaces with the above class of exponent are also obtained.
Sparse pointwise bounds for maximal truncations of rough singular integrals and Sobolev-type inequalities
本文证明粗糙奇异积分极大截断的稀疏逐点控制，导出双权Sobolev不等式及Hedberg型估计，推广至多种加权空间。
Fourier type operators on Orlicz spaces and the role of Orlicz Lebesgue exponents
本文将Fourier乘子、伪微分算子及Fourier积分算子的连续性与波前性质推广到Orlicz空间，并揭示Young函数与相应Lebesgue指数之间的联系。
Interpolation on Convex-set Valued Lebesgue Spaces and its Applications
本文在凸集值Lebesgue空间上证明了两个插值定理，推广了经典插值定理，并据此得到了凸集值分数平均算子与极大算子的有界性及矩阵权重的逆向分解性质。
Optimal function spaces in weighted Sobolev embeddings with $α$-homogeneous weights
本文研究带 $\alpha$-齐次权重的加权 Sobolev 不等式，建立约化原理并刻画最优重排不变目标与定义域空间，通过 Lorentz--Karamata 空间类给出具体例子。
The notion of weakly monotone functions extends the classical definition of monotone function, that can be traced back to H.Lebesgue. It was introduced, in the setting of Sobolev spaces, by J.Manfredi, and thoroughly investigated in the more general framework of Orlicz-Sobolev spaces by diverse authors, including T.Iwaniec, J.Kauhanen, P.Koskela, J.Maly, J.Onninen, X.Zhong. The present paper complements and augments the available theory of pointwise regularity properties of weakly monotone functions in Orlicz-Sobolev spaces. In particular, a variant is proposed in a customary condition ensurin…
We firstly describe a maximal inequality for dual Sobolev spaces W^{-1,p}. This one corresponds to a "Sobolev version" of usual properties of the Hardy-Littlewood maximal operator in Lebesgue spaces. Even in the euclidean space, this one seems to be new and we develop arguments in the general framework of Riemannian manifold. Then we present an application to obtain interpolation results for Sobolev spaces.
For the Hardy-Littlewood maximal and Calderón-Zygmund operators, the weighted boundedness on the Lebesgue spaces are well known. We extend these to the Orlicz-Morrey spaces. Moreover, we prove the weighted boundedness on the weak Orlicz-Morrey spaces. To do this we show the weak-weak modular inequality. The Orlicz-Morrey space and its weak version contain weighted Orlicz, Morrey and Lebesgue spaces and their weak versions as special cases. Then we also get the boundedness for these function spaces as corollaries.
A new general Hormander type condition involving anisotropies and mixed norms is introduced, and boundedness results for Fourier multi- pliers on anisotropic Besov and Triebel-Lizorkin spaces of distributions with mixed Lebesgue norms are obtained. As an application, the continuity of such operators is established on mixed Sobolev and Lebesgue spaces too. Some lifting properties and equivalent norms are obtained as well.
Defect of compactness, relative to an embedding of two Banach spaces E and F, is a difference between a weakly convergent sequence in E and its weak limit taken up to a remainder that vanishes in the norm of F. For Sobolev embeddings in particular, defect of compactness is expressed as a profile decomposition - a sum of terms, called elementary concentrations, with asymptotically disjoint supports. We discuss a profile decomposition for the Sobolev space of a Riemannian manifold with bounded geometry, which is a sum of elementary concentrations associated with concentration profiles defined on…
Space-Time Approximation with Shallow Neural Networks in Fourier Lebesgue spaces
该研究将谱巴伦空间推广到各向异性加权傅里叶-勒贝格空间以适配时空域不同正则性，证明其与博赫纳-索伯列夫空间的包含关系，建立了浅层神经网络在该范数下的逼近速率界，拓展了PDE解的逼近理论。
Defect of compactness, relative to an embedding of two Banach spaces E and F, is a difference between a weakly convergent sequence in E and its weak limit taken up to a remainder that vanishes in the norm of F. For Sobolev embeddings in particular, defect of compactness is expressed as a profile decomposition - a sum of terms, called elementary concentrations, with asymptotically disjoint supports. We discuss a profile decomposition for the Sobolev space of a Riemannian manifold with bounded geometry, which is a sum of elementary concentrations associated with concentration profiles defined on…
Two-weight criteria of various type for the Hardy-Littlewood maximal operator and singular integrals in variable exponent Lebesgue spaces defined on the real line are established.
Motivated by the recent characterization of Sobolev spaces due to Brezis-Van Schaftingen-Yung we prove new weak-type inequalities for one parameter families of operators connected with mixed norm inequalities. The novelty here comes from the fact that the underlying measure space incorporates the parameter as a variable. The connection to classical and fractional order Sobolev spaces is shown through the use of generalized Riesz potential spaces and the Caffarelli-Silvestre extension principle. Higher order inequalities are also considered. We indicate many examples and applications to PDE's a…
Maximal noncompactness of limiting Sobolev embeddings
本文提出一种新方法，用于建立作用于一般拟范数函数空间上的算子的球测度非紧性的下界，并证明Brezis-Wainger型空间中的极限Sobolev嵌入是最大非紧的。
Sobolev type inequalities for fractional maximal functions and Riesz potentials in half spaces
本文研究了半空间中双相泛函加权 Morrey 空间内分数次极大函数与 Riesz 势的 Sobolev 型不等式，并证明了 Riesz 势算子及 Sobolev 函数向加权 Campanato 空间的嵌入性质。
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