Orthogonal bases of Brauer symmetry classes of tensors for groups having cyclic support on non-linear Brauer characters

This paper provides some properties of Brauer symmetry classes of tensors. We derive a dimension formula for the orbital subspaces in the Brauer symmetry classes of tensors corresponding to the irreducible Brauer characters of the groups having cyclic groups support on non-linear Brauer characters. Using the derived formula, we investigate the necessary and sufficient condition for the existence of the o-basis of Dicyclic groups, Semi-dihedral groups and also reinvestigate those things on Dihedral groups. Some criteria for the non-vanishing elements in the Brauer symmetry classes of tensors associated to those groups are also included.Comment: 20 page

A note on a paper of Harris concerning the asymptotic approximation to the eigenvalues of −y"+qy=λy-y"+qy=\lambda y, with boundary conditions of general form

In this paper, we derive an asymptotic approximation to the eigenvalues of the linear differential equation $$-y"(x)+q(x)y(x)=\lambda y(x), x\in (a,b)$$ with boundary conditions of general form, when $q$ is a measurable function which has a singularity in $(a,b)$ and which is integrable on subsets of $(a,b)$ which exclude the singularity

On General multilinear square function with non-smooth kernels

In this paper, we obtain some boundedness of the following general multilinear square functions $T$ with non-smooth kernels, which extend some known results significantly. $$T(\vec{f})(x)=\big( \int_{0}^\infty \big|\int_{(\mathbb{R}^n)^m}K_v(x,y_1,\dots,y_m) \prod_{j=1}^mf_{j}(y_j)dy_1,\dots,dy_m\big|^2\frac{dv}{v}\big)^{\frac 12}.$$ The corresponding multilinear maximal square function $T^*$ was also introduced and weighted strong and weak type estimates for $T^*$ were given.Comment: 19 page

New bounds for bilinear Calder\'on-Zygmund operators and applications

In this work we extend Lacey's domination theorem to prove the pointwise control of bilinear Calder\'on--Zygmund operators with Dini--continuous kernel by sparse operators. The precise bounds are carefully tracked following the spirit in a recent work of Hyt\"onen, Roncal and Tapiola. We also derive new mixed weighted estimates for a general class of bilinear dyadic positive operators using multiple $A_{\infty}$ constants inspired in the Fujii-Wilson and Hrus\v{c}\v{e}v classical constants. These estimates have many new applications including mixed bounds for multilinear Calder\'on--Zygmund operators and their commutators with $BMO$ functions, square functions and multilinear Fourier multipliers.Comment: 35 pages, accepted for publication in Revista Matem\'atica Iberoamerican

Inclusions of Waterman-Shiba spaces into generalized Wiener classes

The characterization of the inclusion of Waterman-Shiba spaces $\Lambda BV^{(p)}$ into generalized Wiener classes of functions $BV(q;\,\delta)$ is given. It uses a new and shorter proof and extends an earlier result of U. Goginava.Comment: 5 page

Pointwise domination and weak L1L^1 boundedness of Littlewood-Paley Operators via sparse operators

In this note, notwithstanding the generalization, we simplify and shorten the proofs of the main results of the third author's paper \cite{SXY} significantly. In particular, the new proof for \cite[Theorem 1.1]{SXY} is quite short and, unlike the original proof, does not rely on the properties of "Marcinkiewicz function". This allows us to get a precise linear dependence on the Dini constants with a subsequent application to Littlewood-Paley operators by the well-known techniques. In other words, we relax the log-Dini condition in the pointwise bound to the classical Dini condition $\int\limits_{0}^{1} \frac{\varphi(t)}{t}dt<\infty$. This proves a well-known open problem (see e.g. \cite[P. 37--38]{CY}).Comment: 31 page

Weighted bounds for multilinear operators with non-smooth kernels

Let $T$ be a multilinear integral operator which is bounded on certain products of Lebesgue spaces on $\mathbb R^n$. We assume that its associated kernel satisfies some mild regularity condition which is weaker than the usual H\"older continuity of those in the class of multilinear Calder\'on-Zygmund singular integral operators. In this paper, given a suitable multiple weight $\vec{w}$, we obtain the bound for the weighted norm of multilinear operators $T$ in terms of $\vec{w}$. As applications, we exploit this result to obtain the weighted bounds for certain singular integral operators such as linear and multilinear Fourier multipliers and the Riesz transforms associated to Schr\"odinger operators on $\mathbb{R}^n$. Our results are new even in the linear case