Bernstein inequality on conic domains and triangle

Author name not available (Why is that?)

Publication date: 3 May 2022

Abstract: We establish weighted Bernstein inequalities in

L^{p}

space for the doubling weight on the conic surface

m a t h b b V_{0}^{d + 1} = (x, t) : | x | = t, x i n m a t h b b R^{d}, t i n [0, 1]

as well as on the solid cone bounded by the conic surface and the hyperplane

t = 1

, which becomes a triangle on the plane when

d = 1

. While the inequalities for the derivatives in the

t

variable behave as expected, there are inequalities for the derivatives in the

x

variables that are stronger than what one may have expected. As an example, on the triangle

(x_{1}, x_{2}) : x_{1} g e 0,, x_{2} g e 0,, x_{1} + x_{2} l e 1

, the usual Bernstein inequality for the derivative

p a r t i a l_{1}

states that

| p h i_{1} p a r t i a l_{1} f |_{p, w} l e c n | f |_{p, w}

with

p h i_{1} (x_{1}, x_{2}) := x_{1} (1 - x_{1} - x_{2})

, whereas our new result gives

| (1-x_2)^{-1/2} phi_1 partial_1 f|_{p,w} le c n |f|_{p,w}.

The new inequality is stronger and points out a phenomenon unobserved hitherto for polygonal domains.

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