Let $L$ be a second order divergence form elliptic operator with complex bounded measurable coefficients. The operators arising in connection with $L$, such as the heat semigroup and Riesz transform, are not, in general, of Calderón-Zygmund type and exhibit behavior different from their counterparts built upon the Laplacian. The current paper aims at a thorough description of the properties of such operators in $L^p$, Sobolev, and some new Hardy spaces naturally associated to $L$. First, we show that the...

We consider a complete connected noncompact Riemannian manifold M with bounded geometry and spectral gap. We prove that the imaginary powers of the Laplacian and the Riesz transform are bounded from the Hardy space X¹(M), introduced in previous work of the authors, to L¹(M).

We consider an abstract non-negative self-adjoint operator L acting on L²(X) which satisfies Davies-Gaffney estimates. Let $H_L^p\left(X\right)$ (p > 0) be the Hardy spaces associated to the operator L. We assume that the doubling condition holds for the metric measure space X. We show that a sharp Hörmander-type spectral multiplier theorem on $H_L^p\left(X\right)$ follows from restriction-type estimates and Davies-Gaffney estimates. We also establish a sharp result for the boundedness of Bochner-Riesz means on $H_L^p\left(X\right)$.

We extend a result of Coifman and Dahlberg [Singular integral characterizations of nonisotropic $H^p$ spaces and the F. and M. Riesz theorem, Proc. Sympos. Pure Math., Vol. 35, pp. 231–234; Amer. Math. Soc., Providence, 1979] on the characterization of $H^p$ spaces by singular integrals of ${ℝ}^n$ with a nonisotropic metric. Then we apply it to produce singular integral versions of generalized BMO spaces. More precisely, if $T_{\lambda }$ is the family of dilations in ${ℝ}^n$ induced by a matrix with a nonnegative eigenvalue, then...

Let ${ℒ}_1=-\Delta +V$ be a Schrödinger operator and let ${ℒ}_2={(-\Delta )}^2+V^2$ be a Schrödinger type operator on ${ℝ}^n$$(n\ge 5)$, where $V\ne 0$ is a nonnegative potential belonging to certain reverse Hölder class...

Four basic results of Marcinkiewicz are presented in summability theory. We show that setting out from these theorems many mathematicians have reached several nice results for trigonometric, Walsh- and Ciesielski-Fourier series.

For 0 < q ≤ 1, the author introduces a new Hardy space $L²H_{ℝ}^q\left(ℝ{²}_{+}\times ℝ{²}_{+}\right)$ on the product domain, and gives its generalized Lusin-area characterization. From this characterization, a φ-transform characterization in M. Frazier and B. Jawerth’s sense is deduced.

The martingale Hardy space $H_p\left({[0,1)}^2\right)$ and the classical Hardy space $H_p{(}^2)$ are introduced. We prove that certain means of the partial sums of the two-parameter Walsh-Fourier and trigonometric-Fourier series are uniformly bounded operators from $H_p$ to $L_p$ (0 < p ≤ 1). As a consequence we obtain strong convergence theorems for the partial sums. The classical Hardy-Littlewood inequality is extended to two-parameter Walsh-Fourier and trigonometric-Fourier coefficients. The dual inequalities are also verified and a...