Singular integrals and fractional powers of operators

Bounds for Singular Fractional Integrals and Related Fourier Integral Operators

Singular Integrals and Littlewood–Paley Operators

We prove mixed Ap-Ar inequalities for several basic singular integrals, Littlewood–Paley operators, and the vector-valued maximal function. Our key point is that r can be taken arbitrarily big. Hence, such inequalities are close in spirit to those obtained recently in the works by T. Hytönen and C. Pérez, and M. Lacey. On one hand, the “Ap-A∞” constant in these works involves two independent su...

Bounds for singular fractional integrals and related Fourier integral operators, preprint

Let Ω ⊂ Ω̃ be open sets in R, I ⊂ R be an open neighborhood of the origin and let η be a compactly supported smooth function on Ω× I; we assume that η(·, 0) does not vanish identically. For each x ∈ Ω let t 7→ Γ(x, t) ⊂ Ω̃ be a regular parametrization of a submanifold Mx ⊂ Ω̃ with codimension l. We assume that Γ(x, t) ⊂ Ω if (x, t) ∈ supp η, and that Γ satisfies Γ(x, 0) = x and depends smoothly on...

Fractional Powers of Operators of Tsallis Ensemble and their Parameter

From this he went on to obtain several other identities in elegant ways which are all central in the development of quantum time evolution, Gibbsian ensembles in equilibrium quantum statistical mechanics, perturbation expansions, inequalities concerning correlation functions etc., all of which depend on the appearance of the exponential operator of the form introduced in Eq.(1). For a comprehen...

Multilinear Singular Operators with Fractional Rank

We prove bounds for multilinear operators on R given by multipliers which are singular along a k dimensional subspace. The new case of interest is when the rank k/d is not an integer. Connections with the concept of true complexity from Additive Combinatorics are also investigated.