Optimal Hölder and $L\sp p$ estimates for $\overline\partial\sb b$ on the boundaries of real ellipsoids in ${\bf C}\sp n$

Shaw, Mei-Chi

doi:10.1090/s0002-9947-1991-1005084-7

Cited by 10 publications

(7 citation statements)

References 26 publications

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“…Proof of Theorem Let ϕ be a (0,1)‐form satisfying the compatibility condition, that is,

\int_{b Ω} ϕ \land α = 0

for every continuous up to the boundary

\bar{\partial}

‐closed (2,0)‐form α on Ω. M.C. Shaw [30, 31] showed that

u : = T_{b} ϕ = H^{+} ϕ - H^{-} ϕ

is an integral solution (in the distribution sense) to the

{\bar{\partial}}_{b}

‐equation,

{\bar{\partial}}_{b} u = ϕ

, on

b Ω

where

H^{+} ϕ false(z false) : = \frac{1}{4 π^{2}} \lim_{ε \to 0^{+}} \int_{b Ω} H false(ζ, z - ε ν (z) false) ϕ false(ζ false) \land ω false(ζ false), H^{-} ϕ false(z false) : = \frac{1}{4 π^{2}} \lim_{ε \to 0^{+}} \int_{b Ω} H false(z + ε ν (z), ζ false) ϕ false(ζ false) \land ω false(ζ false) .

Here

ν (z)

is the outward unit normal vector at

z \in b Ω

;

ω false(ζ false) = d ζ_{1} \land d ζ_{2}

; and

H (ζ, z)

is given by …”

Section: Proof Of Theorem 12 and Theorem 13mentioning

confidence: 99%

“…The purpose of this paper is to prove

L^{p}

estimates,

1 \leq p \leq \infty

, for the tangential Cauchy–Riemann equation

{\bar{\partial}}_{b} u = f

on a class of convex, infinite‐type ellipsoids

Ω \subset {double-struckC}^{2}

. We construct an explicit solution to the

{\bar{\partial}}_{b}

‐equation using the Henkin solution for the Cauchy–Riemann equation and an idea of Shaw [30, 31]. We also present two applications.…”

Section: Introductionmentioning

confidence: 99%

“…The

{\bar{\partial}}_{b}

‐equation has been extensively investigated on classes of domains of finite type including: 1) strongly pseudoconvex domains [13, 20, 21]; 2) convex domains [2, 31]; 3) domains with a diagonalizable Levi form [12]; 4) domains where the Levi form has comparable eigenvalues [23]; 5) decoupled domains [25]; or 6) pseudoconvex domains in

C^{2}

[8, 11]. See also [24, 34].…”

Section: Introductionmentioning

confidence: 99%

“…There are cases, however, where the Blaschke condition is sufficient. Namely, the sufficiency is known when Ω is 1) a strongly pseudoconvex domain [16, 19, 32]; 2) a pseudoconvex domain in

C^{2}

of finite type [5, 30]; 3) a complex or real ellipsoid by [3, 31]; 4) a convex domain of strictly finite type [4, 9, 10] and/or lineally convex and of finite type [6]; 5) an infinite type of domain of the class studied by Ha [17] as well as the example

D_{α} = \{true(z_{1}, z_{2} true) \in C^{2} : {false| z_{1} false|}^{2} + \exp (() 1 + \frac{2}{α} - \frac{1}{false| z_{2} {false|}^{α}}) < 1\}

with

0 < α < 1

(see [1]).…”

Section: Introductionmentioning

confidence: 99%

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Lp‐Estimates for the ∂¯b‐equation on a class of infinite type domains

Khanh

Raich

2020

Mathematische Nachrichten

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We prove estimates, , for solutions to the tangential Cauchy–Riemann equations on a class of infinite type domains . The domains under consideration are a class of convex ellipsoids, and we show that if is a ‐closed (0,1)‐form with coefficients in , then there exists an explicit solution u satisfying . Moreover, when , we show that there is a gain in regularity to an f‐Hölder space.We also present two applications. The first is a solution to the ‐equation, that is, given a smooth (0,1)‐form ϕ on with an L1‐boundary value, we can solve the Cauchy–Riemann equation so that where C is independent of and ϕ. The second application is a discussion of the zero sets of holomorphic functions with zero sets of functions in the Nevanlinna class within our class of domains.

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“…Proof of Theorem Let ϕ be a (0,1)‐form satisfying the compatibility condition, that is,

\int_{b Ω} ϕ \land α = 0

for every continuous up to the boundary

\bar{\partial}

‐closed (2,0)‐form α on Ω. M.C. Shaw [30, 31] showed that

u : = T_{b} ϕ = H^{+} ϕ - H^{-} ϕ

is an integral solution (in the distribution sense) to the

{\bar{\partial}}_{b}

‐equation,

{\bar{\partial}}_{b} u = ϕ

, on

b Ω

where

H^{+} ϕ false(z false) : = \frac{1}{4 π^{2}} \lim_{ε \to 0^{+}} \int_{b Ω} H false(ζ, z - ε ν (z) false) ϕ false(ζ false) \land ω false(ζ false), H^{-} ϕ false(z false) : = \frac{1}{4 π^{2}} \lim_{ε \to 0^{+}} \int_{b Ω} H false(z + ε ν (z), ζ false) ϕ false(ζ false) \land ω false(ζ false) .

Here

ν (z)

is the outward unit normal vector at

z \in b Ω

;

ω false(ζ false) = d ζ_{1} \land d ζ_{2}

; and

H (ζ, z)

is given by …”

Section: Proof Of Theorem 12 and Theorem 13mentioning

confidence: 99%

“…The purpose of this paper is to prove

L^{p}

estimates,

1 \leq p \leq \infty

, for the tangential Cauchy–Riemann equation

{\bar{\partial}}_{b} u = f

on a class of convex, infinite‐type ellipsoids

Ω \subset {double-struckC}^{2}

. We construct an explicit solution to the

{\bar{\partial}}_{b}

‐equation using the Henkin solution for the Cauchy–Riemann equation and an idea of Shaw [30, 31]. We also present two applications.…”

Section: Introductionmentioning

confidence: 99%

“…The

{\bar{\partial}}_{b}

C^{2}

[8, 11]. See also [24, 34].…”

Section: Introductionmentioning

confidence: 99%

“…There are cases, however, where the Blaschke condition is sufficient. Namely, the sufficiency is known when Ω is 1) a strongly pseudoconvex domain [16, 19, 32]; 2) a pseudoconvex domain in

C^{2}

D_{α} = \{true(z_{1}, z_{2} true) \in C^{2} : {false| z_{1} false|}^{2} + \exp (() 1 + \frac{2}{α} - \frac{1}{false| z_{2} {false|}^{α}}) < 1\}

with

0 < α < 1

(see [1]).…”

Section: Introductionmentioning

confidence: 99%

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Lp‐Estimates for the ∂¯b‐equation on a class of infinite type domains

Khanh

Raich

2020

Mathematische Nachrichten

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show abstract

“…Further estimates for ff b are given in [2], [3], [14], [17], [18], [19]. Further estimates for ff b are given in [2], [3], [14], [17], [18], [19].…”

mentioning

confidence: 99%

Local regularity for the tangential Cauchy-Riemann complex.

1993

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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Let G be a strictly pseudoconvex domain in C n with # m -smooth boundary and 0 € bG. By a local homotopy formula for \ we understand the following device. There is given a neighborhood base {M } of relatively open sets M, with 0 G M c £G, and on each M there exist linear integral operators R q (q = l, 2, . . . , n -2), which fulfil the equation (with ^/= 0 for q = n-2) for /e# 0°t9 (M), %/e#£ e + 1 (M). Our aim is to construct homotopy formulae, which satisfy #*-estimatesWe shall also study a slightly more general Situation, where R q fis not necessarily a linear operator. The Standard local solution operators go back to Henkin [5]. These operators are, in a geometrical normalized Situation, the starting point for our investigations. After a local biholomorphic transformation, the surface bG can be assumed strictly convex near 0. So it would suffice to study this case. In order to broaden the possible ränge of applications, we take a slightly more general Situation. After a rotation, bG can be given locally äs the graph over its tangent space at 0 T 0 (bG) = {(z\u)\z f € C""" 1 , weÄP}, z n = u + iv. Now we assume, that we have chosen a strictly convex domain W, resp. W^ => W, in C n ~ i x IR and we are given a function H(z\ u) on W, resp. on ffi, such that M-{(*', u + iH(z\ u))|(2', u)eW} 9 resp. ? over ffi, where H(-9 u) is strictly convex for all fixed u (see chapter l for more details). On ^ and Ü we are given some geometrical and analytical constants 9 which are summarized into one single constant . Brought to you by | University of Iowa Libraries Authenticated Download Date | 6/7/15 10:07 AM Brought to you by | University of Iowa Libraries Authenticated Download Date | 6/7/15 10:07 AM -J ^(/ίΛΑ,,-!^ J ^ f/M/V,-!-*, J «ΐ J *2(^/)AA,,,-l-£ J J 0 +-SxJ 0 +-

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Sharp estimates for $\bar \partial $ on convex domains of finite typeon convex domains of finite type

Cumenge

2001

Ark. Mat.

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Let Q be a bounded convex domain in C n, with smooth boundary of finite type m.The equation c)u~f is solved in f~ with sharp estimates: if f has bounded coefficients, the coefficients of our solution u are in the Lipschitz space A1/m(f~). OptimM estimates are Mso given when data have coefficients belonging to LP([~), p> 1.We solve the c)-equation by means of integral operators whose kernels are not based on the choice of a "good" support function. Weighted kernels are used; in order to reflect the geometry of bf~, we introduce a weight expressed in terms of the Bergman kernel of f/.

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Optimal Hölder and $L\sp p$ estimates for $\overline\partial\sb b$ on the boundaries of real ellipsoids in ${\bf C}\sp n$

Abstract: ABSTRACT. Let D be a real ellipsoid in en, n ~ 3, with defining function p(z) = E~=I (x~nk + y~mk) -1, zk = x k + iYk ,where n k , mk EN. In this paper we study the sharp HOlder and L P estimates for the solutions of the

Cited by 10 publications

References 26 publications

Lp‐Estimates for the ∂¯b‐equation on a class of infinite type domains

Lp‐Estimates for the ∂¯b‐equation on a class of infinite type domains

Local regularity for the tangential Cauchy-Riemann complex.

Sharp estimates for $\bar \partial $ on convex domains of finite typeon convex domains of finite type

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