Estimates of singular integrals and their commutators on weighted Hardy spaces

In this job, let T be a variant of Hörmander’s singular operator and b ∈ Lipβ,ω(ℝn). We adopt the classic Harmonic analysis method and obtain that commutators are bounded on weighted Hardy space and weighted Herz space.

We propose a method for designing accurate interpolation formulas on the real axis for the purpose of function approximation in weighted Hardy spaces. In particular, we consider the Hardy space of functions that are analytic in a strip region around the real axis, being characterized by a weight function $w$ that determines the decay rate of its elements in the neighborhood of infinity. Such a space is considered as a set of functions that are transformed by variable transformations that realize a certain decay rate at infinity. Popular examples of such transformations are given by the single exponential (SE) and double exponential (DE) transformations for the SE-Sinc and DE-Sinc formulas, which are very accurate owing to the accuracy of sinc interpolation in the weighted Hardy spaces with single and double exponential weights $w$, respectively. However, it is not guaranteed that the sinc formulas are optimal in weighted Hardy spaces, although Sugihara has demonstrated that they are near optimal. An explicit form for an optimal approximation formula has only been given in weighted Hardy spaces with SE weights of a certain type. In general cases, explicit forms for optimal formulas have not been provided so far. We adopt a potential theoretic approach to obtain almost optimal formulas in weighted Hardy spaces in the case of general weight functions $w$. We formulate the problem of designing an optimal formula in each space as an optimization problem written in terms of a Green potential with an external field. By solving the optimization problem numerically, we obtain an almost optimal formula in each space. Furthermore, some numerical results demonstrate the validity of this method. In particular, for the case of a DE weight, the formula designed by our method outperforms the DE-Sinc formula.

An application of weighted Hardy spaces to the Navier–Stokes equations

In this paper, we establish the weighted sharp maximal function inequalities for the multilinear operator associated to the singular integral operator with variable Calderon–Zygmund kernel. As an application, we obtain the boundedness of the operator on weighted Lebesgue and Morrey spaces.

Inequalities of the form ${\left \| {{{\left | x \right |}^\alpha }Tf} \right \|_q} \leqq C{\left \| {{{\left | x \right |}^\alpha }f} \right \|_p}$ are proved for certain well-known integral transforms, T, in ${E^n}$. The transforms considered include Calderón-Zygmund singular integrals, singular integrals with variable kernel, fractional integrals and fractional integrals with variable kernel.

In this paper we first study the generalized weighted Hardy spaces $H^{p}_{L,w}(X)$ for $0 < p \le 1$ associated to nonnegative self-adjoint operators $L$ satisfying Gaussian upper bounds on the space of homogeneous type $X$ in both cases of finite and infinite measure. We show that the weighted Hardy spaces defined via maximal functions and atomic decompositions coincide. Then we prove weighted regularity estimates for the Green operators of the inhomogeneous Dirichlet and Neumann problems in suitable bounded or unbounded domains including bounded semiconvex domains, convex regions above a Lipschitz graph and upper half-spaces. Our estimates are in terms of weighted $L^{p}$ spaces for the range $1 < p <\infty$ and in terms of the new weighted Hardy spaces for the range $0 < p \le 1$. Our regularity estimates for the Green operators under the weak smoothness assumptions on the boundaries of the domains are new, especially the estimates on Hardy spaces for the full range $0 < p \le 1$ and the case of unbounded domains.

Navier–Stokes flow in the weighted Hardy space with applications to time decay problem

Let $$(X, d, \mu )$$ be a space of homogeneous type with a metric d and a doubling measure $$\mu $$ . Assume that $$\rho $$ is a critical function on X which has an associated class of weights containing the Muckenhoupt weights as a proper subset. In this paper, we prove the quantitative weighted estimates for certain singular integrals corresponding to the new class of weights. It is important to note that the assumptions on the kernels of these singular integrals do not have any regularity conditions. Our applications include the spectral multipliers and the Riesz transforms associated to Schrodinger operators in various settings, ranging from the magnetic Schrodinger operators in Euclidean spaces to the Laguerre operators.

Let \((X, d, \mu )\) be a metric measure space endowed with a distance \(d\) and a nonnegative Borel doubling measure \(\mu \). Let \(L\) be a second-order non-negative self-adjoint operator on \(L^2(X)\). Assume that the semigroup \(e^{-tL}\) generated by \(L\) satisfies Gaussian upper bounds. In this article we establish a discrete characterization of weighted Hardy spaces \(H_{L, S, w}^{p}(X)\) associated with \(L\) in terms of the area function characterization, and prove its weighted atomic decomposition, where \(0<p\le 1\) and a weight \(w\) is in the Muckenhoupt class \(A_{\infty }\). Further, we introduce a Moser type estimate for \(L\) to show the discrete characterization for the weighted Hardy spaces \(H_{L, G, w}^{p}(X)\) associated with \(L\) in terms of the Littlewood–Paley function and obtain the equivalence between the weighted Hardy spaces in terms of the Littlewood–Paley function and area function.

We consider the spatial numerical range of operators on weighted Hardy spaces and give conditions for closedness of numerical range of compact operators. We also prove that the spatial numerical range of finite rank operators on weighted Hardy spaces is star shaped; though, in general, it does not need to be convex.

The semi-inner product, in the sense of Lumer, on weighted Hardy space which generate the norm is unique. Also we will discuss some properties of the numerical range of bounded linear operators on weighted Hardy spaces.

We establish the mapping properties for some sublinear operators on weighted Hardy spaces with variable exponents by using extrapolation. In particular, we study the Calderón–Zygmund operators, the maximal Bochner–Riesz means, the intrinsic square functions and the Marcinkiewicz integrals on weighted Hardy spaces with variable exponents.

We study the boundedness of Calderon–Zygmund operators on weighted Hardy spaces \(H^p_w\) using Littlewood-Paley theory. It is shown that if a Calderon–Zygmund operator T satisfies T *1 = 0, then T is bounded on \(H^p_w\) for \(w\in A_{p(1+\frac\varepsilon n)}\) and \(\frac n{n+\varepsilon}<p\le1\), where e is the regular exponent of the kernel of T.

Inequalities of the form ‖ | x | α T f ‖ q ≦ C ‖ | x | α f ‖ p {\left \| {{{\left | x \right |}^\alpha }Tf} \right \|_q} \leqq C{\left \| {{{\left | x \right |}^\alpha }f} \right \|_p} are proved for certain well-known integral transforms, T, in E n {E^n} . The transforms considered include Calderón-Zygmund singular integrals, singular integrals with variable kernel, fractional integrals and fractional integrals with variable kernel.

The purpose of this paper is to study the L 2 boundedness of operators of the form f ↦ ψ(x) ∫ f (γ t (x))K(t)dt, where γ t (x) is a C ∞ function defined on a neighborhood of the origin in (t, x) ∈ ℝ N × ℝ n , satisfying γ 0(x) ≡ x, ψ is a C ∞ cut-off function supported on a small neighborhood of 0 ∈ ℝ n , and K is a “multi-parameter singular kernel” supported on a small neighborhood of 0 ∈ ℝ N . The goal is, given an appropriate class of kernels K, to give conditions on γ such that every operator of the above form is bounded on L 2. The case when K is a Calderon-Zygmund kernel was studied by Christ, Nagel, Stein, and Wainger; we generalize their conditions to the case when K has a “multi-parameter” structure. For example, when K is given by a “product kernel.” Even when K is a Calderon- Zygmund kernel, our methods yield some new results. This is the first paper in a three part series, the later two of which are joint with E. M. Stein. The second paper deals with the related question of L p boundedness, while the third paper deals with the special case when γ is real analytic.