• Abstract

In this paper we consider the Heckman-Opdam-Jacobi operator $\Delta_{HJ}$ on $\mathbb R^{d+1}$. We define the Heckman-Opdam-Jacobi intertwining operator $V_{HJ}$, which turn out to be the transmutation operator between $\Delta_{HJ}$ and the Laplacian $\Delta_{d+1}$. Next we construct $^tV_{HJ}$ the dual of this intertwining operator. We exploit these operators to develop a new harmonic analysis corresponding to $\Delta_{HJ}$.

• Abstract

In this paper, we are interested in the regularity estimates of the nonnegative viscosity super solution of the $β$−biased infinity Laplacian equation $$∆^β_∞u = 0,$$ where $β ∈ \mathbb{R}$ is a fixed constant and $∆^β_∞u := ∆^N_∞u + β|Du|,$ which arises from the random game named biased tug-of-war. By studying directly the $β$−biased infinity Laplacian equation, we construct the appropriate exponential cones as barrier functions to establish a key estimate. Based on this estimate, we obtain the Harnack inequality, Hopf boundary point lemma, Lipschitz estimate and the Liouville property etc.

• Abstract

We investigate the initial boundary value problem of some semilinear pseudo-parabolic equations with Newtonian nonlocal term. We establish a lower bound for the blow-up time if blow-up does occur. Also both the upper bound for $T$ and blow up rate of the solution are given when $J(u_0)<0$. Moreover, we establish the blow up result for arbitrary initial energy and the upper bound for $T$. As a product, we refine the lifespan when $J(u_0)<0.$

• Abstract

In the present paper, by extending some fractional calculus to the framework of Clifford analysis, new classes of wavelet functions are presented. Firstly, some classes of monogenic polynomials are provided based on 2-parameters weight functions which extend the classical Jacobi ones in the context of Clifford analysis. The discovered polynomial sets are next applied to introduce new wavelet functions. Reconstruction formula as well as Fourier-Plancherel rules have been proved. The main tool reposes on the extension of fractional derivatives, fractional integrals and fractional Fourier transforms to Clifford analysis.

• Abstract

In this paper, we first introduce some new kinds of weighted amalgam spaces. Then we discuss the strong type and weak type estimates for a class of Calderόn–Zygmund type operators $T_θ$ in these new weighted spaces. Furthermore, the strong type estimate and endpoint estimate of linear commutators $[b, T_θ ]$ formed by $b$ and $T_θ$ are established. Also we study related problems about two-weight, weak type inequalities for $T_θ$ and $[b, T_θ ]$ in the weighted amalgam spaces and give some results.

• Abstract

In this paper, by making use of Divergence theorem for multiple integrals, we establish some integral inequalities for Schur convex functions defined on bodies $B\subset \mathbb{R}^{n}$ that are symmetric, convex and have nonempty interiors. Examples for three dimensional balls are also provided.

• Abstract

Based on the results of the boundedness of $\mu_\Omega^b$ on $L^p$ spaces, by using the theory of atomic decomposition of Hardy spaces, we obtain the boundedness of $\mu_\Omega^b$ on Hardy spaces.

• Abstract

In this paper, we calculate the sharp bound for the generalized $m$-linear $n$-dimensional Hardy-Littlewood-Pόlya operator on power weighted central and noncentral homogeneous Morrey spaces. As an application, the sharp bound for the Hardy-Littlewood-Pόlya operator on power weighted central and noncentral homogeneous Morrey spaces is obtained. Finally, we also find the sharp bound for the Hausdorff operator on power weighted central and noncentral homogeneous Morrey spaces, which generalizes the previous results.

• Abstract

In this paper, we establish the strong and weak boundedness of the multilinear maximal operator in the setting of the Choquet integral with respect to the $\alpha$-dimensional Hausdorff content. Our results cover Orobitg and Verdera's results in [8].

• Abstract

In this paper, we give a brief survey of recent results on axially symmetric Navier-Stokes equations (ASNS) in the following categories: regularity criterion, Liouville property for ancient solutions, decay and vanishing of stationary solutions. Some discussions also touch on the full 3 dimensional equations. Two results, closing of the scaling gap for ASNS and vanishing of homogeneous D solutions in 3 dimensional slabs will be described in more detail.

In the addendum, two new results in the 3rd category will also be presented, which are generalizations of recently published results by the author and coauthors.

• Abstract

The boundedness on Triebel-Lizorkin and Lebesgue spaces of the multilinear operators associated to some singular integral operators satisfying a variant of Hörmander's condition are obtained.

• Abstract

This paper is concerned with the existence of solution for a general class of strongly nonlinear elliptic problems associated with the differential inclusion $$\beta(u)+A(u)+g(x,u,Du)\ni f,$$ where $A$ is a Leray-Lions operator from $W^{1,p}_0(\Omega)$ into its dual, $\beta$ maximal monotone mapping such that $0 \in \beta(0) $, while $g(x,s,\xi)$ is a nonlinear term which has a growth condition with respect to $\xi$ and no  growth with respect to $s$  but it satisfies a sign-condition on $s$. The right hand side $ f$ is assumed to belong to $L^1(\Omega)$.

• Abstract

For the nonlinear degenerate parabolic equations, how to find an appropriate boundary value condition to ensure the well-posedness of weak solution has been an interesting and challenging problem. In this paper, we develop the general characteristic function method to study the stability of weak solutions based on a partial boundary value condition.