In this paper, we study $L^p$-conjecture on locally compact hypergroups and by some technical proofs we give some sufficient and necessary conditions  for a weighted Lebesgue space  $L^p(K,w)$ to be a convolution Banach algebra, where $1<p<infty$, $K$ is a locally compact hypergroup and $w$ is a weight function on $K$.  Among the other things, we also show that if $K$ is a locally compact hyper...

In this paper we investigate the existence and regularity of the solutions to the complex Ginzburg-Landau equation, @ t u = Au + (a + ii))u ? (b + ii)juj 2 u, on the phase space L r;p (R n) of weighted L p functions in innnite domain R n of arbitrary spatial dimensions. The unique local strong solutions are established for subcritical , i.e., r > n p ? 1. Especially, the classical Lebesgue phas...

A weakly continuous near-action of a Polish group G on a standard Lebesgue measure space (X,μ) is whirly if for every A ⊆ X of strictly positive measure and every neighbourhood V of identity in G the set V A has full measure. This is a strong version of ergodicity, and locally compact groups never admit whirly actions. On the contrary, every ergodic near-action by a Polish Lévy group in the sen...

Here #n is the standard Gaussian measure in R, of density d#n(x)=>k=1 ,(xk) dxk , x=(x1 , . . ., xn) # R , ,(xk)=1 2? exp(&xk 2), 8 is the inverse of the distribution function 8 of #1 , and A=[x # R: |x&a|<h for some a # A] denotes the open h-neighborhood of A. (1) becomes identity for all half-spaces A of measure p. In these notes we suggest an equivalent analytic form for (1) involving a rela...

We show that if u is a solution to a linear elliptic differential equation of order 2m ≥ 2 in the half-space with t-independent coefficients, and if u satisfies certain area integral estimates, then the Dirichlet and Neumann boundary values of u exist and lie in a Lebesgue space Lp(Rn) or Sobolev space Ẇ p ±1(R n). Even in the case where u is a solution to a second order equation, our results a...

A criterion for the maximum possible pointwise convergence rate in Birkhoff’s ergodic theorem semiflows a Lebesgue space is obtained. It proved that higher rates of this are impossible.

The work is devoted to the study of Fréchet algebras symmetric (invariant under composition every components its argument with any measure preserving bijection domain argument) analytic functions on Cartesian powers complex Banach spaces Lebesgue integrable in a power $p\in [1,+\infty)$ complex-valued segment $[0,1]$ and semi-axis. We show that algebra all entire bounded type $n$th space $L_p[0...

Let 1 < p ≤ q < +∞ and let v, w be weights on (0,+∞) satisfying: (?) v(x)xis equivalent to a non-decreasing function on (0,+∞) for some ρ ≥ 0; [w(x)x] ≈ [v(x)x] for all x ∈ (0,+∞). We prove that if the averaging operator (Af)(x) := 1 x R x 0 f(t) dt, x ∈ (0,+∞), is bounded from the weighted Lebesgue space Lp((0,+∞); v) into the weighted Lebesgue space Lq((0,+∞);w), then there exists ε0 ∈ (0, p−...

In this paper, we establish the new forms of Riemann-type fractional integral and derivative operators. The novel operator is proved to be bounded in Lebesgue space some classical differential operators are obtained as special cases. properties like semi-group, inverse certain others discussed its weighted Laplace transform evaluated. Fractional integro-differential free-electron laser (FEL) ki...

We study linear control systems in infinite–dimensional Banach spaces governed by analytic semigroups. For p ∈ [1,∞] and α ∈ R we introduce the notion of L–admissibility of type α for unbounded observation and control operators. Generalising earlier work by Le Merdy [20] and the first named author and Le Merdy [12] we give conditions under which L–admissibility of type α is characterised by bou...