We consider weighted Bergman projection Pα : L ∞ (B) → B where α > −1 and B is the Bloch space of the unit ball B of the complex space C n . We obtain the exact norm of the operator Pα where the Bloch space is observed as a space with norm (and semi-norm) induced from the Besov space Bp, 0 < p < ∞, (B∞ = B). Our work contains, as a special case, the main results from [3] and [6]. 1 2010 Mathematics Subject Classification: Primary 30H20, 30H30
In this paper we estimate norms of integral operator induced with Green function related to the Poisson equation in the unit ball with vanishing boundary data.
We prove existence and uniqueness of a solution to the Cauchy problem corresponding to the dynamics capillarity equation $$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t u_{\varepsilon ,\delta } +\mathrm {div} {\mathfrak f}_{\varepsilon ,\delta }(\mathbf{x}, u_{\varepsilon ,\delta })=\varepsilon \Delta u_{\varepsilon ,\delta }+\delta (\varepsilon ) \partial _t \Delta u_{\varepsilon ,\delta }, \ \ \mathbf{x} \in M, \ \ t\ge 0\\ u|_{t=0}=u_0(\mathbf{x}). \end{array}\right. } \end{aligned}$$ ∂ t u ε , δ + div f ε , δ ( x , u ε , δ ) = ε Δ u ε , δ + δ ( ε ) ∂ t Δ u ε , δ , x ∈ M , t ≥ 0 u | t = 0 = u 0 ( x ) . Here, $${{\mathfrak {f}}}_{\varepsilon ,\delta }$$ f ε , δ and $$u_0$$ u 0 are smooth functions while $$\varepsilon $$ ε and $$\delta =\delta (\varepsilon )$$ δ = δ ( ε ) are fixed constants. Assuming $${{\mathfrak {f}}}_{\varepsilon ,\delta } \rightarrow {{\mathfrak {f}}}\in L^p( {\mathbb {R}}^d\times {\mathbb {R}};{\mathbb {R}}^d)$$ f ε , δ → f ∈ L p ( R d × R ; R d ) for some $$1<p<\infty $$ 1 < p < ∞ , strongly as $$\varepsilon \rightarrow 0$$ ε → 0 , we prove that, under an appropriate relationship between $$\varepsilon $$ ε and $$\delta (\varepsilon )$$ δ ( ε ) depending on the regularity of the flux $${{\mathfrak {f}}}$$ f , the sequence of solutions $$(u_{\varepsilon ,\delta })$$ ( u ε , δ ) strongly converges in $$L^1_{loc}({\mathbb {R}}^+\times {\mathbb {R}}^d)$$ L loc 1 ( R + × R d ) toward a solution to the conservation law $$\begin{aligned} \partial _t u +\mathrm {div} {{\mathfrak {f}}}(\mathbf{x}, u)=0. \end{aligned}$$ ∂ t u + div f ( x , u ) = 0 . The main tools employed in the proof are the Leray–Schauder fixed point theorem for the first part and reduction to the kinetic formulation combined with recent results in the velocity averaging theory for the second. These results have the potential to generate a stable semigroup of solutions to the underlying scalar conservation laws different from the Kruzhkov entropy solutions concept.
We estimate the norm of the harmonic Bergman projection in the context of harmonic Besov spaces. We obtain the two-side norm estimates in general L p −case.2010 Mathematics Subject Classification. Primary 46E15,30H25.
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