単位球上におけるtrue biharmonic Bergman kernelに対する下からの評価について

୯Ґٿ্ʹ͓͚Δ true biharmonic Bergman kernel ʹର͢ΔԼ͔ΒͷධՁʹ͍ͭͯ Note on a lower bound estimate for the true biharmonic Bergman kernel over the unit ball

ాதɹਗ਼ت1 Kiyoki Tanaka

Summary

We consider the space of all square integrable biharmonic functions on the unit ball, which is called by the biharmonic Bergman space b2,2₀ (B). We define the the true biharmonic Bergman space b(2),2₀ (B) as b(2),2₀ (B) := b₀2,2(B)  b1,2₀ (B), where b1,2₀ (B) is the harmonic Bergman space. In [10], we obtain properties for true polyharmonic Bergman space. In this paper, based on prop-erties in [10], we give a lower bound estimate for the reproducing kernel of the true biharmonic Bergman space2.

Keywords : polyharmonic Bergman space, true polyharmonic Bergman kernel

1

Introduction

BΛEuclidۭؒ_RN ͷ։୯Ґٿͱ͠,_SΛ_Bͷڥքͱ͢Δɻ_{m ∈ N, α > −1}ʹରͯ͠weighted polyharmonic Bergman space bm,2α (B)Λ

bm,2_α (B) := Hm(B) ∩ L2(B, (1 − |x|2)α_dx)

ͱఆٛ͢Δɻ͜͜Ͱ_{, H}m(B) ͸_B্ͷ polyharmonic functions of degree mશମͷ੒ۭؒ͢ͱ͢ Δɻ͞Βʹ_{, weighted true polyharmonic Bergman space b}(m),2_α (B)Λ

b(m),2_α (B) := bm,2_α (B)  b_αm−1,2(B)(m ≥ 2), b(1),2_α (B) := b1,2_α (B) ͱఆٛ͢Δɻ_bm,2_α (B), b(m),2_α (B)͸࠶ੜ֩HilbertۭؒͰ͋Γ,ͦΕΒͷ࠶ੜ֩ΛͦΕͧΕ_{Rm,α(x, y),} R(m),α(x, y) ͱॻ͘͜ͱʹ͢ΔɻL2(B, (1 − |x|2)αdx) ͔Β b(m),2α (B) ΁ͷorthogonal projection Λ_Q ͱ͢Δͱ͖, Qf (x) = 

BR(m),α(x, y)f (y)(1 − |y|

2₎α_{dy f ∈ L}2_{(B, (1 − |x|}2₎α_dx)

ͱͳΔɻҰൠʹBergmanۭؒ࿦ʹ͓͍ͯ͸, Toeplitz ࡞༻ૉ_Tφ Λ _{orthogonal projection P} Λ

༻͍ͯ _{Tφf = P [φf ]} ͱఆٛ͠,͋Δ_φͷ class ʹ͓͍ͯ operator algebra ߏ଄Λ༩͑Δ(ྫ͑͹

[11], [12] ΛݟΑ)ɻ͜ͷཧ࿦ΛਐΊΔͨΊʹ΋_{, Q}Λද֩͢Ͱ͋Δ_{R(m),α(x, y)} ʹରͯ͠ධՁΛ༩ ͑Δඞཁ͕͋Δɻಛʹඞཁͱ͞ΕΔධՁ͸_{R(m),α(x, y)}ͷ্͔ΒͷධՁͱ_{R(m),α(x, x)}ͷԼ͔Βͷ ධՁͰ͋Δɻྫ͑͹[3]͸harmonic Bergman space on a smooth bounded domain ্ʹఆٛͨ͠

Toeplitz ࡞༻ૉͷಛ௃͚ͮΛ࠶ੜ֩ͷධՁͷΈ͔Β༩͍͑ͯΔɻ

ຊ࿦จͰ͸ _{m = 2}ͷͱ͖_{, unweighted true biharmonic Bergman kernel R(2),0(x, x)}ʹର͢Δ Լ͔ΒͷධՁΛ༩͑Δɻ

1_{େಉେֶڭཆ෦਺ֶڭࣨ}

2_{ຊݚڀ͸,େಉେֶֶ಺ॿ੒੍౓Ͱ͋Δಛผݚڀ঑ྭۚͷॿ੒Λड͚ͨ΋ͷͰ͋Δɻ}

2000 Mathematics Subject Classification. Primary 46E15; Secondary 31B05

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大同大学紀要　第52 巻（2016）

Theorem 1. There exists a constant C > 0 such that R(2),0(x, x) ≥ _{(1− |x|}C _2)N for x ∈ B.

ఆཧʹؔ͢Δ஫ҙͱͯ͠͸_{, harmonic Bergman kernel R1,0(x, x)}ʹରͯ͠͸ධՁ

R1,0(x, x) ≈ _{(1− |x|}C _2)N (1) ͱͳΔ͜ͱ͕஌ΒΕ͓ͯΓ(ྫ͑͹[2]ΛݟΑ), Rm,0(x, x) = R1,0(x, x) + R(2),0(x, x) + · · · + R(m),0(x, x) ͱ࠶ੜ֩ʹର͢ΔҰൠ࿦ΑΓ_{R(m),0(x, x) ≥ 0}Ͱ͋Δ͜ͱ͔Β_{, Rm,0(x, x)}ͷԼ͔ΒͷධՁ Rm,0(x, x) ≥ C(1 − |x|2)−N ͸ಘΒΕ͍ͯΔɻ R(m),0(x, x)ʹର͢ΔԼ͔ΒͷධՁ͸[10]Ͱ༩͑ͨR(m),α(x, y) ͷද͔ࣔΒ͸ѻ͍ͮΒ͍͘͜ͱ ͔ΒಘΒΕ͍ͯͳ͔͕ͬͨ_{, m = 2}ͷͱ͖ͷΈܭࢉʹΑͬͯ_{R(2),0(x, x)}ͷԼ͔ΒͷධՁΛ༩͑Δ ͜ͱ͕Ͱ͖ͨͨΊ,େಉେֶلཁʹ౤ߘ͍ͤͯͨͩ͘͞ɻ

ઌߦݚڀͱͯ͠͸, ۭؒͷߏ଄ͱ͍͏఺ʹ͍ͭͯ͸Ramazanov[8] ͕poly-Bergman spaceͱ Bergman space ͷରԠΛ༩͍͑ͯΔɻ͞Βʹ, Pessoa[7] ͕poly-Bergman space ؒͷରԠΛ Beurling-Ahlfors transform ͱ shift operator Λ߹੒͢Δ͜ͱʹΑͬͯ༩͓͑ͯΓ, ͦΕΛར༻ ͢Δ͜ͱʹΑͬͯ unweighted polyharmonic Bergman space on the unit discͷߏ଄ͱ࠶ੜ֩ͷ

දࣔʹ͍ͭͯݴٴ͍ͯ͠Δɻզʑͷߟ͑ΔۭؒͷఆٛҬ͸࣮_N࣍ݩͰ͋ΔͨΊPessoaͷख๏͸࢖

͑ͳ͍͕,චऀ͸[10]ʹ͓͍࣮ͯ_N࣍ݩͷ։୯ҐٿΛఆٛҬͱ͢Δweighted true polyharmonic Bergman spaceͷਖ਼ن௚ަجఈΛ༩͑_{, true polyharmonic Bergman kernel R(m),0(x, y)}ͷupper estimateΛ༩͍͑ͯΔɻຊ࿦จͰ͸_{, unweighted true biharmonic Bergman kernel R(2),0(x, x)}ʹ ର͢ΔԼ͔ΒͷධՁΛ༩͑ͨͨΊ, true biharmonic Bergman space্Ͱͷ Toeplitz ࡞༻ૉͷ෼ ྨ໰୊Λ࿦͡Δ४උ͕੔ͬͨͱ͍͑Δɻ

2

Calculation of

R

(x, x)

Lemma 2.2. ࡞༻ૉ_{Sm,αf (x) := Δ}m−1_α (1− |x|2)2(m−1)_{f (x)}͸_b1,2_α (B)͔Β_b(m),2_α (B)΁ͷ༗քશ ୯ࣹࣸ૾Ͱ͋Δɻ Lemma 2.2 ͱ(1)͔Β |Sm,0[R1,0(x, ·)](z)|2 =  BR(m),0(z, y)Sm,0[R1,0(x, ·)](y)dy| 2 ≤ R_(m),0(z, ·)2_L2Sm,0[R1,0(x, ·)]2_L2 ≈ R(m),0(z, z)(1 − |x|2)−N ͱͳΔͨΊ, R(m),0(x, x) ≥ C|Sm,0[R1,0(x, ·)](x)|2(1− |x|2)N (2) Λຬͨ͢ఆ਺_{C > 0}͕ଘࡏ͢Δɻಛʹ_{m = 2}ͷͱ͖͸,୯७ܭࢉ͔Β S2,0[R1,0(x, ·)](x) = (−N 2_{+ 10N − 24)|x|}6_{+ (−3N}2_{+ 8N + 16)|x|}4_{+ (−3N}2_{− 2N)|x|}2_{− N}2 |S|(1 − |x|2₎N Λಘͯ,ӈลͷ෼ࢠ͸_{x ∈ B}ͷͱ͖ෛͷఆ਺Ͱ্ʹ༗քͰ͋Δ͜ͱ͔Β, |S2,0[R1,0(x, ·)](x)| ≥ _{(1− |x|}C1_2)N (3) Λຬͨ͢_{C1> 0}͸ଘࡏ͢Δɻ(2)ͱ(3) ΑΓ R(2),0(x, x) ≥ C_{(1− |x|}C1 _2)N 2 (1− |x|2)N ≥ C2 (1− |x|2)N ΛಘΔɻΑͬͯTheorem 1͕ಘΒΕͨɻ

3

Concluding remarks

લઅʹͯ,զʑ͸true biharmonic Bergman kernelʹର͢ΔԼ͔ΒͷධՁΛಘͨɻ_{m ≥ 3} ͷͱ ͖ͷ_{R(m),0(x, x)}ͷධՁٴͼ_{weigthed true polyharmonic Bergman kernel R(m),α(x, x)}ͷධՁ͸ ٕज़্ূ໌Λ༩͑Δ͜ͱ͕Ͱ͖ͳ͔͚ͬͨͩͰಉ༷ͷධՁ͕͔͋ͬͯ͠Δ΂͖Ͱ͋Δɻ༧૝͞Ε ͍ͯΔධՁΛॻ͍͓ͯ͘ͱ, Conjecture R(m),α(x, x) ≈ _{(1− |x|}1_2)N+α Ͱ͋Δɻ ·ͨ, [3]ͱಉ༷ͷٞ࿦Λ͢Δ͜ͱʹΑͬͯ_{, L}2(B, (1 − |x|2)α_dx)͔Β _b(m),2_α (B)΁ͷorthogonal projection Λ༻͍ͯToeplitz࡞༻ૉΛఆٛͨ͠৔߹ͷToeplitz࡞༻ૉͷಛ௃͚ͮ͸ՄೳͰ͋Ζ͏ ͱ༧૝͢Δɻ

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