Hypoellipticity of the Kohn-Laplacian $$\Box _b$$ □ b and of the $$\bar{\partial }$$ ∂ ¯ -Neumann problem by means of subelliptic multipliers

Baracco, Luca; Pinton, Stefano; Zampieri, Giuseppe

doi:10.1007/s00208-014-1144-1

Cited by 12 publications

(19 citation statements)

References 14 publications

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“…The pseudo-resolvent operator is the inverse of T 2 − 2Re(s)T + |s| 2 . We look for conditions on the coefficients a, b, c such that purely imaginary quaternions (s ∈ H with Re(s) = 0) are in the S-resolvent set of T. Since we will prove our results using the Lax-Milgram lemma, we have to define the sesquilinear form associated with  s (T) (the technique related to the Lax-Milgram lemma to solve a differential equation is extensively used in thē-Neumann problem and, in this field, the third author used it in previous studies [16][17][18][19][20][21][22][23] ). So we consider, for…”

Section: The S-spectrum Approach To Fractional Powers and Preliminarymentioning

confidence: 99%

“…Now, consider the second term on the right hand side of (16) and observe that integrating by parts the first and the last term we obtain…”

Section: The S-spectrum Approach To Fractional Powers and Preliminarymentioning

confidence: 99%

“…The case of spectral operators has, for instance, been studied in Gantner, 42 and the theory of groups and semigroups of quaternionic linear operators has been studied in Alpay et al, 43,44 Colombo and Sabadini, 45 and Ghiloni and Recupero. 46 We are now interested in developing ideas from one and several complex variables, such as in previous studies, 16,[47][48][49][50][51][52][53][54][55] to the quaternionic setting.…”

Section: Final Remarksmentioning

confidence: 99%

“…The pseudo‐resolvent operator is the inverse of

T^{2} - 2 Re false(s false) T + false| s |^{2} scriptI .

We look for conditions on the coefficients a , b , c such that purely imaginary quaternions (

s \in double-struckH

with Re( s )=0) are in the S ‐resolvent set of T . Since we will prove our results using the Lax‐Milgram lemma, we have to define the sesquilinear form associated with

Q_{s} false(T false)

(the technique related to the Lax‐Milgram lemma to solve a differential equation is extensively used in the

\overset{\partial}{true}

‐Neumann problem and, in this field, the third author used it in previous studies). So we consider, for u ,

v \in {scriptC}_{c}^{\infty} false(normalΩ, double-struckH false)

{({scriptQ}_{s} (T) u, v)}_{L}^{2} = {(T^{2} u, v)}_{L^{2}} + | s {false|}^{2} {(u, v)}_{L^{2}},

that is,

\int_{normalΩ} \overset{true‾}{{scriptQ}_{s} (T) u} v d μ = \int_{normalΩ} \overset{true‾}{T^{2} u} v d μ + | s {false|}^{2} \int_{normalΩ} \overset{true‾}{u} v d μ,

where dμ = dxdydz .…”

Section: The S‐spectrum Approach To Fractional Powers and Preliminarymentioning

confidence: 99%

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The structure of the fractional powers of the noncommutative Fourier law

Colombo

Peloso

Pinton

2019

Math Methods in App Sciences

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In the recent years, there has been a lot of interest in fractional diffusion and fractional evolution problems. The spectral theory on the S‐spectrum turned out to be an important tool to define new fractional diffusion operators stating from the Fourier law for nonhomogeneous materials. Precisely, let eℓ, eℓ=1,2,3 be orthogonal unit vectors in R3 and let normalΩ⊂R3 be a bounded open set with smooth boundary ∂Ω. Denoting by x_ a point in Ω, the heat equation is obtained replacing the Fourier law given by T=axtrue_∂xe1+bxtrue_∂ye2+cxtrue_∂ze3 into the conservation of energy law. In this paper, we investigate the structure of the fractional powers of the vector operator T, with homogeneous Dirichlet boundary conditions. Recently, we have found sufficient conditions on the coefficients a, b, c:normalΩ→double-struckR such that the fractional powers of T exist in the sense of the S‐spectrum approach. In this paper, we show that under a different set of conditions on the coefficients a, b, c, the fractional powers of T have a different structure.

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Section: The S-spectrum Approach To Fractional Powers and Preliminarymentioning

confidence: 99%

“…Now, consider the second term on the right hand side of (16) and observe that integrating by parts the first and the last term we obtain…”

Section: The S-spectrum Approach To Fractional Powers and Preliminarymentioning

confidence: 99%

Section: Final Remarksmentioning

confidence: 99%

“…The pseudo‐resolvent operator is the inverse of

T^{2} - 2 Re false(s false) T + false| s |^{2} scriptI .

We look for conditions on the coefficients a , b , c such that purely imaginary quaternions (

s \in double-struckH

with Re( s )=0) are in the S ‐resolvent set of T . Since we will prove our results using the Lax‐Milgram lemma, we have to define the sesquilinear form associated with

Q_{s} false(T false)

(the technique related to the Lax‐Milgram lemma to solve a differential equation is extensively used in the

\overset{\partial}{true}

‐Neumann problem and, in this field, the third author used it in previous studies). So we consider, for u ,

v \in {scriptC}_{c}^{\infty} false(normalΩ, double-struckH false)

{({scriptQ}_{s} (T) u, v)}_{L}^{2} = {(T^{2} u, v)}_{L^{2}} + | s {false|}^{2} {(u, v)}_{L^{2}},

that is,

\int_{normalΩ} \overset{true‾}{{scriptQ}_{s} (T) u} v d μ = \int_{normalΩ} \overset{true‾}{T^{2} u} v d μ + | s {false|}^{2} \int_{normalΩ} \overset{true‾}{u} v d μ,

where dμ = dxdydz .…”

Section: The S‐spectrum Approach To Fractional Powers and Preliminarymentioning

confidence: 99%

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The structure of the fractional powers of the noncommutative Fourier law

Colombo

Peloso

Pinton

2019

Math Methods in App Sciences

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“…The most important results in quaternionic operators theory based on the Sspectrum and the associated theory of slice hyperholomorphic functions are contained in the books [3,4,16,17,21,22,25,26], for the case on n-tuples of operators see [23]. (V) Our future research directions will consider the development of ideas from one and several complex variables, such as in [6][7][8][9][28][29][30] to the quaternionic setting.…”

mentioning

confidence: 99%

The Noncommutative Fractional Fourier Law in Bounded and Unbounded Domains

Colombo

González

Pinton

2021

Complex Anal. Oper. Theory

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Using the spectral theory on the S-spectrum it is possible to define the fractional powers of a large class of vector operators. This possibility leads to new fractional diffusion and evolution problems that are of particular interest for nonhomogeneous materials where the Fourier law is not simply the negative gradient operator but it is a nonconstant coefficients differential operator of the form $$\begin{aligned} T=\sum _{\ell =1}^3e_\ell a_\ell (x)\partial _{x_\ell }, \ \ \ x=(x_1,x_2,x_3)\in \overline{\Omega }, \end{aligned}$$ T = ∑ ℓ = 1 3 e ℓ a ℓ ( x ) ∂ x ℓ , x = ( x 1 , x 2 , x 3 ) ∈ Ω ¯ , where, $$\Omega $$ Ω can be either a bounded or an unbounded domain in $$\mathbb {R}^3$$ R 3 whose boundary $$\partial \Omega $$ ∂ Ω is considered suitably regular, $$\overline{\Omega }$$ Ω ¯ is the closure of $$\Omega $$ Ω and $$e_\ell $$ e ℓ , for $$\ell =1,2,3$$ ℓ = 1 , 2 , 3 are the imaginary units of the quaternions $$\mathbb {H}$$ H . The operators $$T_\ell :=a_\ell (x)\partial _{x_\ell }$$ T ℓ : = a ℓ ( x ) ∂ x ℓ , for $$\ell =1,2,3$$ ℓ = 1 , 2 , 3 , are called the components of T and $$a_1$$ a 1 , $$a_2$$ a 2 , $$a_3: \overline{\Omega } \subset \mathbb {R}^3\rightarrow \mathbb {R}$$ a 3 : Ω ¯ ⊂ R 3 → R are the coefficients of T. In this paper we study the generation of the fractional powers of T, denoted by $$P_{\alpha }(T)$$ P α ( T ) for $$\alpha \in (0,1)$$ α ∈ ( 0 , 1 ) , when the operators $$T_\ell $$ T ℓ , for $$\ell =1,2,3$$ ℓ = 1 , 2 , 3 do not commute among themselves. To define the fractional powers $$P_{\alpha }(T)$$ P α ( T ) of T we have to consider the weak formulation of a suitable boundary value problem associated with the pseudo S-resolvent operator of T. In this paper we consider two different boundary conditions. If $$\Omega $$ Ω is unbounded we consider Dirichlet boundary conditions. If $$\Omega $$ Ω is bounded we consider the natural Robin-type boundary conditions associated with the generation of the fractional powers of T represented by the operator $$\sum _{\ell =1}^3a_\ell ^2(x)n_\ell (x) \partial _{x_\ell }+a(x)I$$ ∑ ℓ = 1 3 a ℓ 2 ( x ) n ℓ ( x ) ∂ x ℓ + a ( x ) I , for $$x\in \partial \Omega $$ x ∈ ∂ Ω , where I is the identity operator, $$a:\partial \Omega \rightarrow \mathbb {R}$$ a : ∂ Ω → R is a given function and $$n=(n_1,n_2,n_3)$$ n = ( n 1 , n 2 , n 3 ) is the outward unit normal vector to $$\partial \Omega $$ ∂ Ω . The Robin-type boundary conditions associated with the generation of the fractional powers of T are, in general, different from the Robin boundary conditions associated to the heat diffusion problem which leads to operators of the type $$ \sum _{\ell =1}^3a_\ell (x)n_\ell (x) \partial _{x_\ell }+b(x)I$$ ∑ ℓ = 1 3 a ℓ ( x ) n ℓ ( x ) ∂ x ℓ + b ( x ) I , $$x\in \partial \Omega . $$ x ∈ ∂ Ω . For this reason we also discuss the conditions on the coefficients $$a_1$$ a 1 , $$a_2$$ a 2 , $$a_3: \overline{\Omega } \subset \mathbb {R}^3\rightarrow \mathbb {R}$$ a 3 : Ω ¯ ⊂ R 3 → R of T and on the coefficient $$b:\partial \Omega \rightarrow \mathbb {R}$$ b : ∂ Ω → R so that the fractional powers of T are compatible with the physical Robin boundary conditions for the heat equations.

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Triangular resolutions and effectiveness for holomorphic subelliptic multipliers

Kim

Zaitsev

2021

Advances in Mathematics

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Hypoellipticity of the Kohn-Laplacian $$\Box _b$$ □ b and of the $$\bar{\partial }$$ ∂ ¯ -Neumann problem by means of subelliptic multipliers

Cited by 12 publications

References 14 publications

The structure of the fractional powers of the noncommutative Fourier law

The structure of the fractional powers of the noncommutative Fourier law

The Noncommutative Fractional Fourier Law in Bounded and Unbounded Domains

Triangular resolutions and effectiveness for holomorphic subelliptic multipliers

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