Let H(B) denote the space of all holomorphic functions on the unit ball B ⊂ Cn. Let φ be a holomorphic self-map of B and g ∈ H(B). In this paper, we investigate the boundedness and compactness of the Volterra composition operator

Let D be the open unit disk in the complex plane C, H(D) the class of all analytic functions on D and φ an analytic self-map of D. In order to unify the products of composition, multiplication, and differentiation operators, Stević and Sharma introduced the following so-called Stević-Sharma operator on H(D): Tψ1,ψ2,φf(z) = ψ1(z)f(φ(z)) + ψ2(z)f ′(φ(z)), where ψ1, ψ2 ∈ H(D). By constructing some...

and Applied Analysis 3 Let D be the differentiation operator on H D , that is, Df z f ′ z . For f ∈ H D , the products of composition and differentiation operators DCφ and CφD are defined, respectively, by DCφ ( f ) ( f ◦ φ)′ f ′(φ) φ′, CφD ( f ) f ′ ( φ ) , f ∈ H D . 1.8 The boundedness and compactness of DCφ on the Hardy space were investigated by Hibschweiler and Portnoy in 11 and by Ohno in...

Let g be a holomorphic map of B, where B is the unit ball of C. Let 0 < p < +∞,−n − 1 < q < +∞, q > −1 and α > 0. This paper gives some necessary and sufficient conditions for the Extended Cesáro Operators induced by g to be bounded or compact between generalized Besov space B(p, q) and αBloch space B.

We define and characterize the harmonic Besov space Bp, 1 ≤ p ≤ ∞, on the unit ball B in Rn. We prove that the Besov spaces Bp, 1 ≤ p ≤ ∞, are natural quotient spaces of certain Lp spaces. The dual of Bp, 1 ≤ p < ∞, can be identified with Bq , 1/p + 1/q = 1, and the dual of the little harmonic Bloch space B0 is B1.

Let H(B) denote the space of all holomorphic functions on the unit ball B of Cn . Let α > 0 , f ∈ H(B) with homogeneous expansion f = ∑k=0 fk . The fractional derivative Dα f is defined as Dα f (z) = ∞ ∑ k=0 (k+1)α fk(z). Let φ be a holomorphic self-map of B and g ∈ H(B) such that g(0) = 0 . In this paper we consider the following integral-type operator

Let H(D) denote the space of all holomorphic functions on the unit disk D of C. Let φ be a holomorphic self-map of D, n be a positive integer and g ∈ H(D). In this paper, we investigate the boundedness and compactness of a generalized integration operator

We reduce here end-point estimates for one singular operator (namely for dyadic square function) to Monge–Ampère equations with drift. The spaces are weighted spaces, and therefore the domain, where we solve our PDE is non-convex. If we are in the end-point situation our goal is either to find a logarithmic blow-up of the norm estimate from below, or to prove that there is upper estimate of the...

In this paper, we show a parabolic version of the Ogawa type inequality in Sobolev spaces. Our inequality provides an estimate of the L∞ norm of a function in terms of its parabolic BMO norm, with the aid of the square root of the logarithmic dependency of a higher order Sobolev norm. The proof is mainly based on the Littlewood-Paley decomposition and a characterization of parabolic BMO spaces....