Optimal in tail Hölder estimates for weak solutions of the nonlocal parabolic p-Laplace equations on the Heisenberg group

Abstract: We prove the Hölder continuity for weak solutions to parabolic p-Laplace equations on the Heisenberg group. We deduce this result while considering an optimal tail condition.

• The paper proves local Hölder continuity for weak solutions in both singular and degenerate regimes using intrinsic scaling, De Giorgi iteration, and expansion of positivity.
• The explicit exponent includes the term εsp/[2(1+ε)], quantitatively showing how L^{1+ε}-in-time tail integrability controls the Hölder modulus.
• The result adapts Euclidean nonlocal regularity methods to Heisenberg geometry while leaving local boundedness for p≤2Q/(Q+2s) and nonsymmetric kernels as open problems.

Setting and main result

This paper establishes local Hölder continuity of weak solutions to the nonlocal parabolic $p$-Laplace equation

$$\partial_t u(x,t) + \text{p.v.}\int_{\mathbb{H}^N} K(x,y,t)\,|u(x,t)-u(y,t)|^{p-2}(u(x,t)-u(y,t))\,dy = 0$$

on $\Omega_T = \Omega\times(0,T]$, where $\Omega$ is a bounded open subset of the Heisenberg group $\mathbb{H}^N$ and the kernel satisfies the standard ellipticity bounds

$$\frac{\lambda'}{|y^{-1}\circ x|_{\mathbb{H}}^{Q+sp}} \leq K(x,y,t) \leq \frac{\Lambda'}{|y^{-1}\circ x|_{\mathbb{H}}^{Q+sp}}, \qquad Q = 2N+2.$$

The central contribution is that the Hölder estimate is obtained under an optimal tail condition: rather than requiring a locally bounded tail ($L^\infty_{\text{loc}}$ in time), the author assumes only

$$\int_{\mathbb{H}^N}\frac{|u(x,\cdot)|^{p-1}}{1+|x|_{\mathbb{H}}^{Q+sp}}\,dx \in L^{1+\varepsilon}_{\text{loc}}(0,T]$$

for some $\varepsilon>0$. The main theorem states that any locally bounded weak solution is locally Hölder continuous, with an explicit modulus: for cylinders $(x_0,t_0)+\mathcal{Q}^\ominus_R(\omega^{2-p})$, the essential oscillation obeys

$$\partial_t u(x,t) + \text{p.v.}\int_{\mathbb{H}^N} K(x,y,t)\,|u(x,t)-u(y,t)|^{p-2}(u(x,t)-u(y,t))\,dy = 0$$0

where the Hölder exponent is

$$\partial_t u(x,t) + \text{p.v.}\int_{\mathbb{H}^N} K(x,y,t)\,|u(x,t)-u(y,t)|^{p-2}(u(x,t)-u(y,t))\,dy = 0$$1

and the quantity $$\partial_t u(x,t) + \text{p.v.}\int_{\mathbb{H}^N} K(x,y,t)\,|u(x,t)-u(y,t)|^{p-2}(u(x,t)-u(y,t))\,dy = 0$$2 incorporates both the local $$\partial_t u(x,t) + \text{p.v.}\int_{\mathbb{H}^N} K(x,y,t)\,|u(x,t)-u(y,t)|^{p-2}(u(x,t)-u(y,t))\,dy = 0$$3 norm and the $$\partial_t u(x,t) + \text{p.v.}\int_{\mathbb{H}^N} K(x,y,t)\,|u(x,t)-u(y,t)|^{p-2}(u(x,t)-u(y,t))\,dy = 0$$4-in-time tail. The third term in the minimum shows precisely how the integrability exponent of the tail enters the modulus — this is where the optimal tail assumption pays off quantitatively.

The result is new for nonlocal parabolic equations on the Heisenberg group, and it extends to the subelliptic setting the program initiated by Kassmann–Weidner [KW24_25] for nonlocal heat equations and completed by Liao [Lia24_5] for Euclidean nonlocal parabolic $$\partial_t u(x,t) + \text{p.v.}\int_{\mathbb{H}^N} K(x,y,t)\,|u(x,t)-u(y,t)|^{p-2}(u(x,t)-u(y,t))\,dy = 0$$5-Laplace equations. It also builds directly on the author's companion work with Tewary [KT25_1], which established local boundedness under the same optimal tail condition for $$\partial_t u(x,t) + \text{p.v.}\int_{\mathbb{H}^N} K(x,y,t)\,|u(x,t)-u(y,t)|^{p-2}(u(x,t)-u(y,t))\,dy = 0$$6.

Methodological framework

The proof follows the intrinsic-scaling (De Giorgi–DiBenedetto) paradigm adapted to the nonlocal setting, closely modeled on Liao's approach [Lia24_5]. Two structural features distinguish it from earlier treatments such as Ding–Zhang–Zhou [DZZ21] and Adimurthi–Prasad–Tewary [APT22_22]:

• No logarithmic estimates or exponential changes of variables are used. In the singular case $$\partial_t u(x,t) + \text{p.v.}\int_{\mathbb{H}^N} K(x,y,t)\,|u(x,t)-u(y,t)|^{p-2}(u(x,t)-u(y,t))\,dy = 0$$7, the "good term" present in the Caccioppoli inequality plays the role that logarithmic estimates play elsewhere; in the degenerate case $$\partial_t u(x,t) + \text{p.v.}\int_{\mathbb{H}^N} K(x,y,t)\,|u(x,t)-u(y,t)|^{p-2}(u(x,t)-u(y,t))\,dy = 0$$8, expansion of positivity replaces the exponential change of variables.
• Tail control via iteration: because the long-range behavior of the solution cannot be discarded, every De Giorgi-type lemma carries an either-or alternative involving the parabolic tail

$$\partial_t u(x,t) + \text{p.v.}\int_{\mathbb{H}^N} K(x,y,t)\,|u(x,t)-u(y,t)|^{p-2}(u(x,t)-u(y,t))\,dy = 0$$9

and the induction must show these alternatives never dominate the oscillation reduction.

A technical device worth noting is the modified truncation level: the energy estimate is applied not to $\Omega_T = \Omega\times(0,T]$0 but to $\Omega_T = \Omega\times(0,T]$1, where $\Omega_T = \Omega\times(0,T]$2 absorbs the far-field tail contribution. This allows the tail terms generated inside the cylinder to cancel against the explicit $\Omega_T = \Omega\times(0,T]$3-weighted tail term in the Caccioppoli inequality, leaving only genuinely far-field contributions to be controlled by the global tail hypothesis.

Core lemmas

The machinery consists of four De Giorgi-type results, all derived from the Caccioppoli inequality:

• De Giorgi lemma (backward in time): if the superlevel set $\Omega_T = \Omega\times(0,T]$4 occupies at most a fraction $\Omega_T = \Omega\times(0,T]$5 of a backward cylinder, then either the tail exceeds $\Omega_T = \Omega\times(0,T]$6 or the solution stays above/below $\Omega_T = \Omega\times(0,T]$7 on the half-radius cylinder. The proof runs a shrinking-balls iteration using a parabolic Sobolev embedding (Proposition 1.7) with exponent $\Omega_T = \Omega\times(0,T]$8 when $\Omega_T = \Omega\times(0,T]$9 and $\Omega$0 otherwise.
• De Giorgi lemma forward in time: the analogous statement from pointwise initial data, proved with time-independent cutoffs so that no time-derivative penalty appears.
• Measure-theoretic propagation forward in time: a set occupying measure $\Omega$1 at time $\Omega$2 retains measure $\Omega$3 (at a reduced level $\Omega$4) throughout a time interval of length $\Omega$5, with $\Omega$6.
• Shrinking lemma: converts sustained measure-theoretic information into smallness of the complementary superlevel sets, using the good term in the energy estimate — the author notes explicitly that no De Giorgi isoperimetric inequality is available in the nonlocal setting due to jumps in $\Omega$7 functions, and that Cozzi's workaround [Coz17_6] via the good term is adopted here.

These combine into the expansion of positivity (Lemma 4.1) for $\Omega$8: positivity on a fraction $\Omega$9 of $\mathbb{H}^N$0 at one time level propagates to the full ball $\mathbb{H}^N$1 over a later time interval, with constants $\mathbb{H}^N$2 and $\mathbb{H}^N$3.

The singular case $\mathbb{H}^N$4

The oscillation reduction proceeds by induction on nested cylinders $\mathbb{H}^N$5 with $\mathbb{H}^N$6. At each step, expansion of positivity yields

$\mathbb{H}^N$7

and the crux is showing the tail term does not destroy the reduction. Splitting the tail integral into contributions from previously visited annuli and the far field, and using the induction hierarchy $\mathbb{H}^N$8, the author obtains

$\mathbb{H}^N$9

which is bounded by $$\frac{\lambda'}{|y^{-1}\circ x|_{\mathbb{H}}^{Q+sp}} \leq K(x,y,t) \leq \frac{\Lambda'}{|y^{-1}\circ x|_{\mathbb{H}}^{Q+sp}}, \qquad Q = 2N+2.$$0 provided $$\frac{\lambda'}{|y^{-1}\circ x|_{\mathbb{H}}^{Q+sp}} \leq K(x,y,t) \leq \frac{\Lambda'}{|y^{-1}\circ x|_{\mathbb{H}}^{Q+sp}}, \qquad Q = 2N+2.$$1 — a geometric-series condition forcing the spatial scaling factor $$\frac{\lambda'}{|y^{-1}\circ x|_{\mathbb{H}}^{Q+sp}} \leq K(x,y,t) \leq \frac{\Lambda'}{|y^{-1}\circ x|_{\mathbb{H}}^{Q+sp}}, \qquad Q = 2N+2.$$2 to be small enough relative to the oscillation contraction factor. This closes the induction with $$\frac{\lambda'}{|y^{-1}\circ x|_{\mathbb{H}}^{Q+sp}} \leq K(x,y,t) \leq \frac{\Lambda'}{|y^{-1}\circ x|_{\mathbb{H}}^{Q+sp}}, \qquad Q = 2N+2.$$3 for a data-dependent constant, yielding the modulus with exponent $$\frac{\lambda'}{|y^{-1}\circ x|_{\mathbb{H}}^{Q+sp}} \leq K(x,y,t) \leq \frac{\Lambda'}{|y^{-1}\circ x|_{\mathbb{H}}^{Q+sp}}, \qquad Q = 2N+2.$$4 plus an additive far-field tail term.

The degenerate case $$\frac{\lambda'}{|y^{-1}\circ x|_{\mathbb{H}}^{Q+sp}} \leq K(x,y,t) \leq \frac{\Lambda'}{|y^{-1}\circ x|_{\mathbb{H}}^{Q+sp}}, \qquad Q = 2N+2.$$5

For $$\frac{\lambda'}{|y^{-1}\circ x|_{\mathbb{H}}^{Q+sp}} \leq K(x,y,t) \leq \frac{\Lambda'}{|y^{-1}\circ x|_{\mathbb{H}}^{Q+sp}}, \qquad Q = 2N+2.$$6 the structure changes: the intrinsic time scale $$\frac{\lambda'}{|y^{-1}\circ x|_{\mathbb{H}}^{Q+sp}} \leq K(x,y,t) \leq \frac{\Lambda'}{|y^{-1}\circ x|_{\mathbb{H}}^{Q+sp}}, \qquad Q = 2N+2.$$7 shrinks as the oscillation decreases, and the argument splits into two alternatives following Liao's scheme [Lia24_2]. In the first alternative (supersolution near its infimum), the De Giorgi lemma and its forward-in-time version produce a pointwise lower bound reducing the oscillation by a factor $$\frac{\lambda'}{|y^{-1}\circ x|_{\mathbb{H}}^{Q+sp}} \leq K(x,y,t) \leq \frac{\Lambda'}{|y^{-1}\circ x|_{\mathbb{H}}^{Q+sp}}, \qquad Q = 2N+2.$$8. In the second alternative (subsolution near its supremum), measure information at a well-chosen time slice $$\frac{\lambda'}{|y^{-1}\circ x|_{\mathbb{H}}^{Q+sp}} \leq K(x,y,t) \leq \frac{\Lambda'}{|y^{-1}\circ x|_{\mathbb{H}}^{Q+sp}}, \qquad Q = 2N+2.$$9 is propagated forward via Lemma 3.3, then fed through the shrinking lemma and the backward De Giorgi lemma to obtain a reduction by $L^\infty_{\text{loc}}$0.

An additional parameter $L^\infty_{\text{loc}}$1 stretches the cylinder height, determined by the constraint $L^\infty_{\text{loc}}$2, and the final choice of $L^\infty_{\text{loc}}$3 must satisfy two conditions simultaneously — the geometric-series bound and $L^\infty_{\text{loc}}$4 — while $L^\infty_{\text{loc}}$5 is taken smaller than both the inclusion constraint and $L^\infty_{\text{loc}}$6. The resulting modulus has exponent $L^\infty_{\text{loc}}$7, which differs from the singular-case exponent because the cylinder heights now contract at rate $L^\infty_{\text{loc}}$8 rather than $L^\infty_{\text{loc}}$9.

Both cases yield a unified estimate with the intermediate radius $$\int_{\mathbb{H}^N}\frac{|u(x,\cdot)|^{p-1}}{1+|x|_{\mathbb{H}}^{Q+sp}}\,dx \in L^{1+\varepsilon}_{\text{loc}}(0,T]$$0, giving the general continuity modulus (GenCont) used in the final theorem.

Completion under the optimal tail hypothesis

The final step converts the additive far-field tail term into a power of $$\int_{\mathbb{H}^N}\frac{|u(x,\cdot)|^{p-1}}{1+|x|_{\mathbb{H}}^{Q+sp}}\,dx \in L^{1+\varepsilon}_{\text{loc}}(0,T]$$1 using the $$\int_{\mathbb{H}^N}\frac{|u(x,\cdot)|^{p-1}}{1+|x|_{\mathbb{H}}^{Q+sp}}\,dx \in L^{1+\varepsilon}_{\text{loc}}(0,T]$$2-in-time assumption. Hölder's inequality gives

$$\int_{\mathbb{H}^N}\frac{|u(x,\cdot)|^{p-1}}{1+|x|_{\mathbb{H}}^{Q+sp}}\,dx \in L^{1+\varepsilon}_{\text{loc}}(0,T]$$3

which produces exactly the third term in the definition of $$\int_{\mathbb{H}^N}\frac{|u(x,\cdot)|^{p-1}}{1+|x|_{\mathbb{H}}^{Q+sp}}\,dx \in L^{1+\varepsilon}_{\text{loc}}(0,T]$$4. This is the quantitative payoff of the optimal tail condition: weaker assumptions (e.g., mere $$\int_{\mathbb{H}^N}\frac{|u(x,\cdot)|^{p-1}}{1+|x|_{\mathbb{H}}^{Q+sp}}\,dx \in L^{1+\varepsilon}_{\text{loc}}(0,T]$$5 tails) would leave a non-vanishing additive term and no Hölder modulus.

It should be noted that the theorem takes local boundedness of $$\int_{\mathbb{H}^N}\frac{|u(x,\cdot)|^{p-1}}{1+|x|_{\mathbb{H}}^{Q+sp}}\,dx \in L^{1+\varepsilon}_{\text{loc}}(0,T]$$6 as a hypothesis. As the author remarks, local boundedness was proved in [KT25_1] only for $$\int_{\mathbb{H}^N}\frac{|u(x,\cdot)|^{p-1}}{1+|x|_{\mathbb{H}}^{Q+sp}}\,dx \in L^{1+\varepsilon}_{\text{loc}}(0,T]$$7; for $$\int_{\mathbb{H}^N}\frac{|u(x,\cdot)|^{p-1}}{1+|x|_{\mathbb{H}}^{Q+sp}}\,dx \in L^{1+\varepsilon}_{\text{loc}}(0,T]$$8 it is expected but not established here that weak solutions are locally bounded under additional integrability, and the boundary value $$\int_{\mathbb{H}^N}\frac{|u(x,\cdot)|^{p-1}}{1+|x|_{\mathbb{H}}^{Q+sp}}\,dx \in L^{1+\varepsilon}_{\text{loc}}(0,T]$$9 plays no role in the Hölder argument itself.

Limitations and open questions

Several restrictions are inherent to the result as stated. First, the solution is assumed locally bounded; extending local boundedness to the full range $\varepsilon>0$0 under the optimal tail condition remains open, and would be needed to make the theorem unconditional for $\varepsilon>0$1. Second, the kernel is assumed symmetric and homogeneous of degree $\varepsilon>0$2; whether the Hölder estimate extends to nonsymmetric kernels on $\varepsilon>0$3, in the spirit of Kassmann–Weidner's Euclidean work [KW25_23], is not addressed. Third, the constants and the exponent $\varepsilon>0$4 depend on the structural data $\varepsilon>0$5 and on $\varepsilon>0$6; no claim of optimality of $\varepsilon>0$7 itself is made beyond the optimality of the tail integrability assumption. Finally, the paper treats only bounded kernels with exact homogeneity; kernels with more general growth or measurable coefficients in the subelliptic setting remain outside the scope.

Conclusion

The paper proves local Hölder continuity for weak solutions of nonlocal parabolic $\varepsilon>0$8-Laplace equations on the Heisenberg group under the optimal $\varepsilon>0$9-in-time tail condition, covering both the singular ($(x_0,t_0)+\mathcal{Q}^\ominus_R(\omega^{2-p})$0) and degenerate ($(x_0,t_0)+\mathcal{Q}^\ominus_R(\omega^{2-p})$1) regimes with an explicit Hölder exponent that quantifies the role of tail integrability. The proof adapts the intrinsic-scaling machinery of Liao's Euclidean theory to the subelliptic geometry of $(x_0,t_0)+\mathcal{Q}^\ominus_R(\omega^{2-p})$2, replacing logarithmic estimates and exponential changes of variables with the good term in the Caccioppoli inequality and expansion of positivity respectively, and managing the long-range interaction through a careful iterative control of tail terms across the induction. The main open problem left by the paper is the removal of the local boundedness hypothesis for $(x_0,t_0)+\mathcal{Q}^\ominus_R(\omega^{2-p})$3.