Weighted Norm Inequalities Research Articles

Weighted norm inequalities for jump and variation of some singular integral operators with rough kernel

Weighted norm inequalities for jump and variation of some singular integral operators with rough kernel

Two weighted norm inequalities for generalized one-sided maximal function on one-sided weighted like Morrey space

Two weighted norm inequalities for generalized one-sided maximal function on one-sided weighted like Morrey space

Weighted norm inequalities for derivatives on Bergman spaces

Weighted norm inequalities for derivatives on Bergman spaces

Weighted norm inequalities, embedding theorems and integration operators on vector-valued Fock spaces

Weighted norm inequalities, embedding theorems and integration operators on vector-valued Fock spaces

Weighted norm inequalities for integral transforms with splitting kernels

We obtain necessary and sufficient conditions on weights for a wide class of integral transforms to be bounded between weighted $L^p-L^q$ spaces, with $1\leq p\leq q\leq \infty$. The kernels $K(x,y)$ of such transforms are only assumed to satisfy upper bounds given by products of two functions, one in each variable. The obtained results are applicable to a number of transforms, some of which are included here as particular examples. Some of the new results derived here are the characterization of weights for the boundedness of the $\mathscr{H}_\alpha$ (or Struve) transform in the case $\alpha>\frac{1}{2}$, or the characterization of power weights for which the Laplace transform is bounded in the limiting cases $p=1$ or $q=\infty$.

Bilinear sparse domination for oscillatory integral operators

In this paper, we prove bilinear sparse domination bounds for a wide class of Fourier integral operators of general rank, as well as oscillatory integral operators associated to Hörmander symbol classes Sρ,δm\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$S^m_{\\rho ,\\delta }$$\\end{document} for all 0≤ρ≤1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$0\\le \\rho \\le 1$$\\end{document} and 0≤δ<1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$0\\le \\delta < 1$$\\end{document}, a notable example is the Schrödinger operator. As a consequence, one obtains weak (1, 1) estimates, vector-valued estimates, and a wide range of weighted norm inequalities for these classes of operators.

A note on Rubio de Francia's extrapolation in tent spaces and applications

The Rubio de Francia extrapolation theorem is a very powerful result which states that in order to show that certain operators satisfy weighted norm inequalities with Muckenhoupt weights it suffices to see that the corresponding inequalities hold for some fixed exponent, for instance p=2. In this paper we extend this result and show that this extrapolation principle allows one to obtain weighted estimates in tent spaces. From our extrapolation result we automatically derive new estimates (and reprove some other) concerning Calderón-Zygmund operators, operators associated with the Kato conjecture, or fractional operators.

Weighted norm inequalities for Schrödinger operators on variable Lebesgue spaces

Weighted norm inequalities for Schrödinger operators on variable Lebesgue spaces

Minimal [formula omitted]-solutions to singular sublinear elliptic problems

We solve the existence problem for the minimal positive solutions u∈Lp(Ω,dx) to the Dirichlet problems for sublinear elliptic equations of the form Lu=σuq+μinΩ,lim infx→yu(x)=0y∈∂∞Ω,where 0<q<1 and Lu≔−div(A(x)∇u) is a linear uniformly elliptic operator with bounded measurable coefficients. The coefficient σ and data μ are nonnegative Radon measures on an arbitrary domain Ω⊂Rn with a positive Green function associated with L. Our techniques are based on the use of sharp Green potential pointwise estimates, weighted norm inequalities, and norm estimates in terms of generalized energy.

Weighted norm inequalities related to fractional Schrödinger operators

Weighted norm inequalities related to fractional Schrödinger operators

Two Weighted Norm Inequalities of Potential Type Operator on Herz Spaces

Two Weighted Norm Inequalities of Potential Type Operator on Herz Spaces

Multiple Weighted Norm Inequalities for Multilinear Strongly Singular Integral Operators with Generalized Kernels

Multiple Weighted Norm Inequalities for Multilinear Strongly Singular Integral Operators with Generalized Kernels

Weighted norm inequalities for multilinear strongly singular Calderón–Zygmund operators on RD‐spaces

AbstractLet be an RD‐space, namely, a space of homogeneous type in the sense of Coifman and Weiss with the Borel measure μ satisfying the reverse doubling condition on . Based on this space, the authors define a multilinear strongly singular Calderón–Zygmund operator whose kernel does not need any size condition and has more singularities near the diagonal than that of a standard multilinear Calderón–Zygmund operator. For such an operator, we establish its boundedness on product of weighted Lebesgue spaces by means of the pointwise estimate for the sharp maximal function. In addition, the endpoint estimates of the type are also obtained. Moreover, we prove weighted boundedness results for multilinear commutators generated by multilinear strongly singular Calderón–Zygmund operators and BMO functions. These results contribute to the extension of multilinear strongly singular Calderón–Zygmund operator theory in the Euclidean case to the context of space of homogeneous type.

Weighted Norm Inequalities for Calderón–Zygmund Operators of $$\phi$$-Type and Their Commutators

Weighted Norm Inequalities for Calderón–Zygmund Operators of $$\phi$$-Type and Their Commutators

Weighted norm inequalities for one sided vector valued Hardy-Littlewood maximal function on one sided Morrey like spaces

Weighted norm inequalities for one sided vector valued Hardy-Littlewood maximal function on one sided Morrey like spaces

Weak and strong type estimates for square functions associated with operators

Let L be a linear operator on L2(Rn) which generates a semigroup e−tL whose kernels pt(x,y) satisfy the Gaussian upper bound. In this paper, we investigate several kinds of weighted norm inequalities for the conical square function Sα,L associated with an abstract operator L. We first establish two-weight inequalities including bump estimates, and Fefferman-Stein inequalities with arbitrary weights. We also present the local decay estimates using the extrapolation techniques, and the mixed weak type estimates corresponding to Sawyer's conjecture by means of a Coifman-Fefferman inequality. Beyond that, we consider other weak type estimates including the restricted weak-type (p,p) for Sα,L and the endpoint estimate for commutators of Sα,L. Finally, all the conclusions aforementioned can be applied to a number of square functions associated to L.

A class of multilinear bounded oscillation operators on measure spaces and applications

In this paper, we develop a comprehensive weighted theory for a class of Banach-valued multilinear bounded oscillation operators on measure spaces, which merges multilinear Calder\'{o}n-Zygmund operators with a quantity of operators beyond the multilinear Calder\'{o}n-Zygmund theory. We prove that such multilinear operators and corresponding commutators are locally pointwise dominated by two sparse dyadic operators, respectively. We also establish three kinds of typical estimates: local exponential decay estimates, mixed weak type estimates, and sharp weighted norm inequalities. Beyond that, based on Rubio de Francia extrapolation for abstract multilinear compact operators, we obtain weighted compactness for commutators of specific multilinear operators on spaces of homogeneous type. A compact extrapolation allows us to get full range of exponents, while weighted interpolation for multilinear compact operators is crucial to the compact extrapolation. These are due to a weighted Fr\'{e}chet-Kolmogorov theorem in the quasi-Banach range, which gives a characterization of relative compactness of subsets in weighted Lebesgue spaces. As applications, we illustrate multilinear bounded oscillation operators with examples including multilinear Hardy-Littlewood maximal operators on measure spaces, multilinear $\omega$-Calder\'{o}n-Zygmund operators on spaces of homogeneous type, multilinear Littlewood-Paley square operators, multilinear Fourier integral operators, higher order Calder\'{o}n commutators, maximally modulated multilinear singular integrals, and $q$-variation of $\omega$-Calder\'{o}n-Zygmund operators.

New weighted norm inequalities for multilinear singular integral operators with generalized kernels and their commutators

New weighted norm inequalities for multilinear singular integral operators with generalized kernels and their commutators

Sparse bounds for the bilinear spherical maximal function

AbstractWe derive sparse bounds for the bilinear spherical maximal function in any dimension . When , this immediately recovers the sharp bound of the operator and implies quantitative weighted norm inequalities with respect to bilinear Muckenhoupt weights, which seems to be the first of their kind for the operator. The key innovation is a group of newly developed continuity improving estimates for the single‐scale bilinear spherical averaging operator.

Weighted Characterization of Parabolic Fractional Maximal Operator

In this paper, we introduce the parabolic off-diagonal Muckenhoupt A + weights. We reveal the properties of these weights, and our main results characterize them through weak-type and strong-type weighted norm inequalities for the fractional forward-in-time maximal operator. Moreover, our results cover some well-known results.