We show that for quasi-greedy bases in real or complex Banach spaces the error of the thresholding greedy algorithm of order N is bounded by the best N term error of approximation times a function of N which depends on the democracy functions and the quasi-greedy constant of the basis. If the basis is democratic this function is bounded by C logN . We show with two examples that this bound is a...

Let [Formula: see text] be a generalized Calderón-Zygmund operator or [Formula: see text] ( the identity operator), let [Formula: see text] and [Formula: see text] be the linear operators, and let [Formula: see text]. Denote the Toeplitz type operator by [Formula: see text]where [Formula: see text] and [Formula: see text] is fractional integral operator. In this paper, we establish the sharp ma...

The problem of the boundedness of the fractional maximal operator Mα, 0 ≤ α < n in local Morrey-type spaces is reduced to the problem of the boundedness of the Hardy operator in weighted Lp-spaces on the cone of non-negative non-increasing functions. This allows obtaining sharp sufficient conditions for the boundedness for all admissible values of the parameters.

We derive uniform weighted L 2 and Morrey-Campanato type estimates for Helmholtz Equations in a medium with a variable index which is not necessarily constant at innnity. Our technique is based on a multiplier method with appropriate weights which generalize those of Morawetz for the wave equation. We also extend our method to the wave equation.

Let H be a complex Hilbert space. Denote by B(H)+ the set of all positive bounded linear operators on H. A bijective map φ : B(H)+ → B(H)+ is said to preserve Lebesgue decompositions in both directions if for any quadruple A,B,C,D of positive operators, B = C +D is an A-Lebesgue decomposition of B if and only if φ(B) = φ(C)+φ(D) is a φ(A)-Lebesgue decomposition of φ(B). It is proved that every ...

Recall that a function u is harmonic (superharmonic, subharmonic) in an open set U ⊂ Rn (n ≥ 1) if u ∈ C2(U) and Δu = 0 (Δu ≤ 0,Δu ≥ 0) on U . Denote by H(U) the space of harmonic functions in U and SH(U) (sH(U)) the subset of C2(U) consisting of superharmonic (subharmonic) functions in U . If A ⊂ Rn is Lebesgue measurable, L1(A) denotes the space of Lebesgue integrable functions on A and |A| d...