Optimal semigroup estimates and functional calculus for the Laplacian on weighted Sobolev spaces

Abstract: In this paper, we consider the Laplace operator on the half-space with Dirichlet and Neumann boundary conditions. These operators are studied on Sobolev spaces with power weights measuring the distance to the boundary. We prove optimal estimates for the resolvent operators and the corresponding heat semigroups. In addition, it is proved that the Dirichlet and Neumann Laplacians admit a bounded $H<sup>\infty$-functional calculus on Sobolev spaces with certain compatibility conditions at the boundary. We show that these compatibility conditions cannot be omitted in general. The results in this paper are a direct extension of those obtained by Lindemulder, Lorist, the author, and Veraar in [J. Funct. Anal., 289(8):110985, 2025].

• The paper establishes sharp two-sided polynomial growth rates for Dirichlet and Neumann heat semigroups, extending the valid weight range to all γ > −1 and removing earlier ε-losses.
• It proves that shifted Laplacians are sectorial of angle zero and have bounded H∞ functional calculus on UMD-valued weighted Sobolev spaces, using density, resolvent estimates, and similarity arguments.
• The results show boundary compatibility conditions are necessary for sectoriality in the low-weight regime, while critical exponents and logarithmic growth remain open cases.

Setting and problem

This paper studies the Dirichlet and Neumann Laplacians $\Delta_{\mathrm{D}}$ and $\Delta_{\mathrm{N}}$ on the half-space $\mathbb{R}^d_+ = (0,\infty)\times\mathbb{R}^{d-1}$, acting on $X$-valued Sobolev spaces $W^{k,p}(\mathbb{R}^d_+, w_\gamma; X)$ with power weights $w_\gamma(x) = |x_1|^\gamma$ measuring distance to the boundary, where $p\in(1,\infty)$, $k\in\mathbb{N}_0$, $\gamma>-1$, and $X$ is a UMD Banach space. The motivation comes from (stochastic) PDE theory, where power weights damp irregular behavior near the boundary, and bounded functional calculus is central to the analysis of linear parabolic SPDEs.

The work extends and sharpens results of Lindemulder, Lorist, Roodenburg, and Veraar (J. Funct. Anal. 2025). In the earlier work it was shown that on certain weighted Sobolev spaces the Dirichlet and Neumann Laplacians are not sectorial and generate analytic $\Delta_{\mathrm{N}}$0-semigroups with polynomial growth, with only almost-optimal growth estimates for a restricted range of weight exponents. The present paper closes this gap: it establishes the exact optimal growth rate, extends the admissible range of weight exponents to all $\Delta_{\mathrm{N}}$1, and proves bounded $\Delta_{\mathrm{N}}$2-functional calculus for the shifted operators, together with the necessity of the compatibility conditions at the boundary.

Main theorem

The paper's central result concerns the realizations $\Delta_{\mathrm{N}}$3, $\Delta_{\mathrm{N}}$4, on weighted Sobolev spaces with compatibility conditions. Writing $\Delta_{\mathrm{N}}$5 and $\Delta_{\mathrm{N}}$6, the space $\Delta_{\mathrm{N}}$7 consists of all $\Delta_{\mathrm{N}}$8 whose traces $\Delta_{\mathrm{N}}$9 vanish whenever $\mathbb{R}^d_+ = (0,\infty)\times\mathbb{R}^{d-1}$0; these are precisely the traces that are well defined at this smoothness and weight. For $\mathbb{R}^d_+ = (0,\infty)\times\mathbb{R}^{d-1}$1 above the threshold $\mathbb{R}^d_+ = (0,\infty)\times\mathbb{R}^{d-1}$2 no compatibility conditions are needed, while in the intermediate regime $\mathbb{R}^d_+ = (0,\infty)\times\mathbb{R}^{d-1}$3 only the single condition $\mathbb{R}^d_+ = (0,\infty)\times\mathbb{R}^{d-1}$4 is imposed.

Under the assumption $\mathbb{R}^d_+ = (0,\infty)\times\mathbb{R}^{d-1}$5, with $\mathbb{R}^d_+ = (0,\infty)\times\mathbb{R}^{d-1}$6 UMD, and with domain $\mathbb{R}^d_+ = (0,\infty)\times\mathbb{R}^{d-1}$7, the main theorem asserts:

• For every $\mathbb{R}^d_+ = (0,\infty)\times\mathbb{R}^{d-1}$8, the operator $\mathbb{R}^d_+ = (0,\infty)\times\mathbb{R}^{d-1}$9 is sectorial of angle zero.
• $X$0, the Dirichlet/Neumann heat semigroup obtained by odd/even reflection of the Gaussian kernel, is an analytic $X$1-semigroup generated by $X$2.
• For every $X$3, $X$4 admits a bounded $X$5-functional calculus of angle zero.
• If $X$6, then $X$7 is uniformly bounded, and sectoriality and the bounded calculus hold already at $X$8.
• If $X$9, then $W^{k,p}(\mathbb{R}^d_+, w_\gamma; X)$0 has exact polynomial growth:

$W^{k,p}(\mathbb{R}^d_+, w_\gamma; X)$1

This improves the earlier upper bound $W^{k,p}(\mathbb{R}^d_+, w_\gamma; X)$2 (valid for $W^{k,p}(\mathbb{R}^d_+, w_\gamma; X)$3) to the sharp rate $W^{k,p}(\mathbb{R}^d_+, w_\gamma; X)$4, and extends the admissible weight exponents from the interval $W^{k,p}(\mathbb{R}^d_+, w_\gamma; X)$5 down to all $W^{k,p}(\mathbb{R}^d_+, w_\gamma; X)$6. The exclusion of the critical values $W^{k,p}(\mathbb{R}^d_+, w_\gamma; X)$7 is a genuine limitation of the method: there the trace characterization of the generator domain fails. The upper endpoint $W^{k,p}(\mathbb{R}^d_+, w_\gamma; X)$8 is optimal, since for larger $W^{k,p}(\mathbb{R}^d_+, w_\gamma; X)$9 the semigroup is not well defined on the space.

Sectoriality and functional calculus

The proof of the calculus results proceeds through three ingredients. First, a density result shows that the space of test functions $w_\gamma(x) = |x_1|^\gamma$0 — smooth functions vanishing to the required even/odd order at the boundary — is dense in $w_\gamma(x) = |x_1|^\gamma$1, where $w_\gamma(x) = |x_1|^\gamma$2 is the largest trace order that exists. The proof combines a known density theorem with a surjectivity result for the trace operator onto a product of weighted Besov spaces, together with a polynomial-correction cut-off argument.

Second, elliptic regularity for the resolvent equation is obtained by extending the right-hand side to all of $w_\gamma(x) = |x_1|^\gamma$3 via the odd (Dirichlet) or even (Neumann) reflection and solving on $w_\gamma(x) = |x_1|^\gamma$4 by the Fourier multiplier $w_\gamma(x) = |x_1|^\gamma$5. For general $w_\gamma(x) = |x_1|^\gamma$6, the estimate

$w_\gamma(x) = |x_1|^\gamma$7

is derived by differentiating the equation to reduce to the case of no compatibility conditions, applying the earlier results at a lower smoothness level, and closing with a density argument. The constant here depends on $w_\gamma(x) = |x_1|^\gamma$8 and blows up as $w_\gamma(x) = |x_1|^\gamma$9; the precise blow-up rate is the subject of the one-dimensional analysis. A consequence is the domain characterization $p\in(1,\infty)$0 with equivalent norms for all $p\in(1,\infty)$1 (and the analogous statement at smoothness $p\in(1,\infty)$2 for the Dirichlet case).

Third, boundedness of the $p\in(1,\infty)$3-calculus for $p\in(1,\infty)$4 is obtained by a similarity argument: choosing $p\in(1,\infty)$5, the lower-order realization $p\in(1,\infty)$6 on $p\in(1,\infty)$7 carries no compatibility conditions and hence has a bounded calculus by the earlier work; since $p\in(1,\infty)$8 and $p\in(1,\infty)$9 with $k\in\mathbb{N}_0$0, the calculus transfers via $k\in\mathbb{N}_0$1.

Optimal one-dimensional resolvent estimates

The sharp semigroup growth rates are obtained from sharp resolvent estimates in one dimension, using the equivalence (valid for operators whose resolvent satisfies a polynomial blow-up bound) between resolvent estimates of the form $k\in\mathbb{N}_0$2 and semigroup bounds $k\in\mathbb{N}_0$3 on sectors.

For large weight exponents $k\in\mathbb{N}_0$4, the resolvent is estimated via the explicit kernel $k\in\mathbb{N}_0$5 with the bound

$k\in\mathbb{N}_0$6

combined with a weighted Hardy-type inequality whose proof is given in detail. The resulting estimate involves the function $k\in\mathbb{N}_0$7, which captures the exact blow-up as $k\in\mathbb{N}_0$8; the derivative estimates for intermediate orders follow from the resolvent equation and interpolation. A logarithmic variant of the Hardy inequality at the critical exponent $k\in\mathbb{N}_0$9 is recorded for later use.

For general weight exponents $\gamma>-1$0, the small-$\gamma>-1$1 estimates follow from the large-exponent case applied at reduced smoothness $\gamma>-1$2, and the large-$\gamma>-1$3 estimates are proved via an explicit commutator formula expressing $\gamma>-1$4 in terms of the resolvent applied to $\gamma>-1$5 plus explicit boundary terms of the form $\gamma>-1$6 times traces of $\gamma>-1$7. On the space with compatibility conditions the boundary terms vanish, yielding the uniform estimate

$\gamma>-1$8

while on the larger space $\gamma>-1$9 without compatibility conditions one obtains the factor $X$0 with $X$1. Crucially, this factor is shown to be sharp: for the test function $X$2 the reverse triangle inequality gives $X$3 for large $X$4. Combining both regimes yields the one-dimensional resolvent estimate with the constant $X$5, which is exactly the form required for the optimal semigroup growth.

Semigroup growth and necessity of compatibility

For $X$6, the resolvent estimates and the abstract equivalence immediately give the upper bound $X$7 on sectors. The matching lower bound is obtained by testing against $X$8, for which $X$9 for $\Delta_{\mathrm{N}}$00; the semigroup property supplies the additive constant $\Delta_{\mathrm{N}}$01 since the semigroup is unbounded. The extension to $\Delta_{\mathrm{N}}$02 uses the factorization $\Delta_{\mathrm{N}}$03 together with the decomposition of $\Delta_{\mathrm{N}}$04 as an intersection of mixed-norm spaces; the tangential heat semigroup $\Delta_{\mathrm{N}}$05 is uniformly bounded on $\Delta_{\mathrm{N}}$06 for any Banach space $\Delta_{\mathrm{N}}$07, and the lower bound is produced by tensor products with scaled Schwartz functions. The same equivalence in the reverse direction yields resolvent estimates in arbitrary dimension.

A notable negative result establishes the necessity of the compatibility conditions: if $\Delta_{\mathrm{N}}$08, then on $\Delta_{\mathrm{N}}$09 without compatibility conditions the realization $\Delta_{\mathrm{N}}$10 is not sectorial for any $\Delta_{\mathrm{N}}$11. The proof combines the sharpness of the factor $\Delta_{\mathrm{N}}$12 with the resolvent identity $\Delta_{\mathrm{N}}$13, so the resolvent norm grows without bound along $\Delta_{\mathrm{N}}$14. This extends the classical unweighted counterexamples of Desch–Dorensmaier to the weighted setting, and it clarifies the trade-off: for fixed $\Delta_{\mathrm{N}}$15, fewer compatibility conditions are required as $\Delta_{\mathrm{N}}$16 increases, and none are needed in the regime studied previously.

Limitations and open questions

The paper leaves several points open. The critical values $\Delta_{\mathrm{N}}$17 are excluded because the domain characterization of the generator fails; for $\Delta_{\mathrm{N}}$18 the author expects the growth estimates to persist, with the generator domain identified as a closure of test functions. At the critical exponent $\Delta_{\mathrm{N}}$19 the growth rate is not established, but a conjecture is stated with a proof sketch: $\Delta_{\mathrm{N}}$20, based on the logarithmic Hardy estimate for the upper bound and an explicit test-function construction for the lower bound. Finally, the polynomial blow-up is attributed to the growth of the weight $\Delta_{\mathrm{N}}$21 at infinity rather than to the boundary behavior: for bounded domains the semigroup is bounded, and for weights $\Delta_{\mathrm{N}}$22 that agree with $\Delta_{\mathrm{N}}$23 near the boundary but are constant for $\Delta_{\mathrm{N}}$24, the author expects uniform boundedness of the semigroup for all admissible parameters; a proof strategy via a split Hardy estimate is indicated, though the result is stated as expected rather than proved.

Conclusion

The paper completes the program initiated in the predecessor work by establishing optimal, two-sided polynomial growth rates for the Dirichlet and Neumann heat semigroups on power-weighted Sobolev spaces over the half-space, extending the admissible weight exponents to all $\Delta_{\mathrm{N}}$25, and proving that the shifted operators $\Delta_{\mathrm{N}}$26 have bounded $\Delta_{\mathrm{N}}$27-functional calculus of angle zero on UMD-valued spaces. The compatibility conditions at the boundary are shown to be both sufficient and, in the low-weight regime, necessary for sectoriality. The main technical advances are the sharp one-dimensional resolvent estimates via explicit kernels and commutator identities, and the transfer of these estimates to arbitrary dimension through the normal–tangential factorization of the semigroup.